DOCSLIB.ORG

Ch 6-Propagation in Periodic Media

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ch 6-Propagation in Periodic Media Bloch waves and Bandgaps Chapter 6 Physics 208, Electro-optics Peter Beyersdorf Document info Ch 6, 1 Bloch Waves There are various classes of boundary conditions for which solutions to the wave equation are not plane waves Planar conductor results in standing waves E(z) = 2E0 sin (kzz) cos (ωt) Waveguide and cavities results in modal structure ikz E(x, y, z) = Enm(x, y)e− Periodic materials result in Bloch waves 2π/Λ iK" "r E! (!r) = E(K! , !r)e− · dK! !0 Ch 6, 2 Class Outline Types of periodic media dispersion relation in layered materials Bragg reflection Coupled mode theory Surface waves Ch 6, 3 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 4 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 5 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 6 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 7 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 8 Bloch’s Theorem 2π/Λ Λ iK" "r E! (!r) = E(K! , !r)e− · dK! !0 Wave solutions in a periodic medium (Bloch waves) are different than in a homogenous medium (plane waves) A correction factor E(K,r) accounts for the difference between plane wave solutions and Bloch wave solutions Wave amplitude has a periodicity defined by the underlying medium, Ek(K,r)=Ek(K,r+Λ) E(K,r) are normal modes of propagation Ch 6, 9 Waves in Layered Media For a wave normally incident on an Λ isotropic layered material, we’ll find the “dispersion relationship” (ω vs K curve) for Bloch waves. This will tell us about the behavior of waves in the material. We’ll see the periodic structure reflects certain wavelengths. This is referred to as Bragg reflection. Ch 6,10 Wave Equation in Layered Media Starting with the wave equation ∂2E! ! ! E! + µ" = 0 ∇ × ∇ × ∂t2 ! " we will plug in the dielectric tensor written as a Fourier series with periodicity Λ i 2πl z !(z) = !le− Λ !l and an arbitrary wave i(kz+ωt) E! = E!0(k)e− dk to get ! 2 ikz 2 i 2πl z ikz k E!0(k)e− dk + ω µ #le− Λ E!0(k)e− dk = 0 Ch 6,11 ! "l ! Wave Equation in Layered Media 2πl 2 ! ikz 2 i Λ z ! ikz with k E0(k)e− dk + ω µ #le− E0(k)e− dk = 0 ! "l ! 2πl k! = k + and defining Λ 2 ikz 2 2πl ik!z gives k E! (k)e− dk ω µ # E! (k" )e− dk" = 0 0 − l 0 − Λ ! ! "l or 2 2 2πl ikz k E!0(k) ω µ #lE!0(k ) e− dk = 0 " − − Λ $ ! #l so 2πl k2E! (k) ω2µ # E! (k ) = 0 for all k 0 − l 0 − Λ !l Ch 6,12 Wave Equation in Layered Media for all k Is an infinite set of equations. Consider equations for: kΛ/2π=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3 … K=0 coupled K=0.2π/Λ K=0.4π/Λ 2 Let K be the value value of k±2πl/Λ closest to ω με0 in a series of coupled equations, where ε0 is the zeroth order Fourier coefficient of ε(z). The whole series of equations for -∞<k<∞ can be treated instead as a Λ series of coupled equations for 0<K<2π/ . T2heπl solution to each set of k2E! (k) ω2µ # E! (k ) = 0 equations for a value of K0 onl−y containsl 0 te−rmsΛ at k=K±2πl/Λ, thus !l i(kz+ωt) 2π i(K l 2π )z iωt E! (z) = E!0(k)e dk E! (K, z) = E!0(K l )e − Λ − → − Λ Ch 6,13 ! !l Bloch Waves in Layered Media The Bloch waves are normal modes of propagation so 2π/Λ iKz E! (z) = E(K, z)e− dK !0 and each mode is composed of plane wave components of amplitude E(K±2πl/Λ). To find these amplitudes we consider the dispersion relation 2πl Since this rekp2rE!esen(k) tsω2 µan infin# E! (kite se)t= of0 coupled 0 − l 0 − Λ equations, we will exam!inl e this expression and isolate the equations that couple most strongly to E0(K), ignore the rest and solve for E0(k) Ch 6,14 Field Components in Layered Media 2 2 2 2π 2 2π k E!0(k) ω µ#øE!0(k) ω µ#1E!0(k ) ω µ# 1E!0(k + ) . = 0 or − − − Λ − − Λ − Allowing us to express the E0(K-2πl/Λ) amplitudes as 1 2 2π 2 2π E!0(K) = ω µ#1E!0(K ) + ω µ# 1E!0(K + ) . K2 ω2µ# − Λ − Λ − − ø ! " ! 2π 1 2 ! 2π 2 ! E0(K ) = 2 ω µ$1E0(K 2 ) + ω µ$ 1E0(K) . − Λ K 2π ω2µ$ − Λ − − − Λ − ø # $ ! 2 ! " 2 ! 2πl ! 2π k E0(k)1 ω µ #lE20(k ! ) = 02 ! 2π E0(K + ) = 2− ω µ$1E−0(ΛK) + ω µ$ 1E0(K + 2 ) . Λ K + 2π ω2µ$l − Λ − Λ − !ø # $ ! " … Ch 6,15 Resonant Coupling of Waves Momentum of forwards wave k with wavenumber k is ℏk, for -k backwards wave it is -ℏk. kg Grating can be thought of as superposition of forwards and backwards going waves, with momenta ±ℏkg, where kg=2π/Λ. For light to couple between forwards and backwards waves, momentum must be conserved ℏk+mℏkg=-ℏk This is like a collision of a forward photon with m phonons producing a backwards photon Ch 6,16 Field Components in Layered Media 2 ! 1 22π ! 2 2π 2 ! 2π In theE 0c(aKse) = whe2 re 2##K # lω#µ#1#!E0ω#(Kµ##ø i.e). +theω µ #fo1rEwa0(Krd+ wa)ve . K ω µ#ø− Λ − Λ − Λ − momentum canno−t be! co!nve"rted to the backward wave " ! 2π 1 2 ! 2π 2 ! momEen0(Ktum b)y= the addit2ion of a ωkicµk$1 Efr0(oKm the2 )g+raωtinµ$g 1inE 0(theK) . − Λ K 2π ω2µ$ − Λ − − layered material, o−nlyΛ the− l=0ø t#erm is significant and the $ 2 ! 2 ! 2πl dispersion 2rπelatio! n K E"10(K) ω µ 2 #lE0(K 2) = 0 2π E!0(K + ) = − ω µ$1E!0(K−) +Λω µ$ 1E!0(K + 2 ) . Λ 2π 2 2 l − Λ − for any value of KK +is Λuncouωplµe$ød# t!o that for other values of K, $ 2 2 − and gives##K # ω# µ#"ø =# 0 meaning the phase velocity is that due − ! " to the average index of refraction for the medium Ch 6,17 Coupling of Field Components ! 2πl 1 2 ! 2π(l 1) 2 ! 2πl E0(K + ) = 2 ω µ$1E0(K + − ) + ω µ$ 1E0(K + 2 ) . Λ K + 2πl ω2µ$ Λ − Λ − Λ − ø # $ l=0 ! " l=-1 l=+1 ! 1 2 ! 2π 2 ! 2π E0(K) = 2 2 ω µ#1E0(K ) + ω µ# 1E0(K + ) . K ω µ#ø − Λ − Λ − − ! 2 " 2π 2 In the case2π where# K# 1 l# # # ω #µ#ø2 for some n2oπn-ze2ro value of E!0(K ) = − Λ ≈ ω µ$1E!0(K 2 ) + ω µ$ 1E!0(K) . Λ ! 2π 2 " 2 Λ − l=m this −l=m termK is also sigω µn$øificant and f−or the dispersion − − Λ − # $ relation 2 ! 2 ! 2πl 2π ! K E"10(K) ω µ 2 #lE0(K 2) = 0 2π E!0(K + ) = − ω µ$1E!0(K−) +Λω µ$ 1E!0(K + 2 ) . Λ 2π 2 2 l − Λ − K + ω µ$ø ! Λ − # $ ! " Λ we need only consider two values of K, i.e. K and K-2!m/ . Ch 6,18 Dispersion Relation in Layered Media The dispersion relation Considering only terms with E(K) and E(K-2πm/Λ) gives 2πm K2 ω2µ" E# (K) ω2µ" E# (K ) = 0 − ø 0 − m 0 − Λ 2! " 2πm 2 2πm 2 K ω µ"ø E#0 K ω µ" mE#0(K) = 0 − Λ − − Λ − − #$ % & $ % A nontrivial solution to these coupled equations only exists if 2 2 2 K ω µ"ø ω µ"m −2 2−πm 2 2 = 0 2 ω µ" m 2 K Λ 2πωl µ"ø k! E−!0(k) − ω µ #−lE!0(k − ) = 0! ! − − Λ ! Λ ! "l # ! 1 i2πmz/Λ and for ε-m=εm*! in a lossless! medium, since!!m = !(z)e− dz 2 Λ 0 2πm 2! K2 ω2µ" K ω2µ" ω2µ " = 0 − ø − Λ − ø − | m| Ch 6,19 #$ % & ! " ! " Dispersion Relation in Layered Media which can be solved for K, the Bloch wave vector for a wave of frequency ω 2 2 mπ mπ 2 2 2 2πm 2 K = ! + "øµω ( "m µω ) + "øµω Λ ± " Λ ± & | | Λ "$ % ' ( # 2 2πm 2 K2 ω2µ" K ω2µ" ω2µ " = 0 − ø − Λ − ø − | m| Ch 6,20 graph of disp#e$rsion rel%ationsh&ip (figure 6.2) ! " ! " Bandgaps in Layered Media 2 When the Bragg condition is met (K-2πm/Λ=ω ε0 μ) K2 K2 ω2 < ω2 > real solutions exist for# # #µ (#" +# " # ) # and µ (" " ) ø | m| ø − | m| 2 2 mπ mπ 2 2 2 22 2πm 2 K = ! K+ "øµω 2 ( "m µωK) + "øµω SolutionsΛ f±or" Λ <±ω&<| | Λ "$ µ (%!ø + !m ) µ (!ø !m') ( # | | − | | are complex, this region is called the forbidden band.
Load more

Recommended publications
  • Double Negative Dispersion Relations from Coated Plasmonic Rods∗
    MULTISCALE MODEL. SIMUL. c 2013 Society for Industrial and Applied Mathematics Vol. 11, No. 1, pp. 192–212 DOUBLE NEGATIVE DISPERSION RELATIONS FROM COATED PLASMONIC RODS∗ YUE CHEN† AND ROBERT LIPTON‡ Abstract. A metamaterial with frequency dependent double negative effective properties is constructed from a subwavelength periodic array of coated rods. Explicit power series are developed for the dispersion relation and associated Bloch wave solutions. The expansion parameter is the ratio of the length scale of the periodic lattice to the wavelength. Direct numerical simulations for finite size period cells show that the leading order term in the power series for the dispersion relation is a good predictor of the dispersive behavior of the metamaterial. Key words. metamaterials, dispersion relations, Bloch waves, simulations AMS subject classifications. 35Q60, 68U20, 78A48, 78M40 DOI. 10.1137/120864702 1. Introduction. Metamaterials are artificial materials designed to have elec- tromagnetic properties not generally found in nature. One contemporary area of research explores novel subwavelength constructions that deliver metamaterials with both a negative bulk dielectric constant and bulk magnetic permeability across cer- tain frequency intervals. These double negative materials are promising materials for the creation of negative index superlenses that overcome the small diffraction limit and have great potential in applications such as biomedical imaging, optical lithography, and data storage. The early work of Veselago [39] identified novel ef- fects associated with hypothetical materials for which both the dielectric constant and magnetic permeability are simultaneously negative. Such double negative media support electromagnetic wave propagation in which the phase velocity is antiparallel to the direction of energy flow and other unusual electromagnetic effects, such as the reversal of the Doppler effect and Cerenkov radiation.
    [Show full text]
  • Plasma Waves
    Plasma Waves S.M.Lea January 2007 1 General considerations To consider the different possible normal modes of a plasma, we will usually begin by assuming that there is an equilibrium in which the plasma parameters such as density and magnetic field are uniform and constant in time. We will then look at small perturbations away from this equilibrium, and investigate the time and space dependence of those perturbations. The usual notation is to label the equilibrium quantities with a subscript 0, e.g. n0, and the pertrubed quantities with a subscript 1, eg n1. Then the assumption of small perturbations is n /n 1. When the perturbations are small, we can generally ignore j 1 0j ¿ squares and higher powers of these quantities, thus obtaining a set of linear equations for the unknowns. These linear equations may be Fourier transformed in both space and time, thus reducing the differential equations to a set of algebraic equations. Equivalently, we may assume that each perturbed quantity has the mathematical form n = n exp i~k ~x iωt (1) 1 ¢ ¡ where the real part is implicitly assumed. Th³is form descri´bes a wave. The amplitude n is in ~ general complex, allowing for a non•zero phase constant φ0. The vector k, called the wave vector, gives both the direction of propagation of the wave and the wavelength: k = 2π/λ; ω is the angular frequency. There is a relation between ω and ~k that is determined by the physical properties of the system. The function ω ~k is called the dispersion relation for the wave.
    [Show full text]
  • Dispersion Relation & Index Ellipsoids
    8/27/2020 Advanced Electromagnetics: 21st Century Electromagnetics Dispersion Relation & Index Ellipsoids Lecture Outline • Dispersion relation • Dispersion surfaces • Index ellipsoids Slide 2 1 8/27/2020 Dispersion Relation Slide 3 The Wave Vector The wave vector (wave momentum) is a vector quantity that conveys two pieces of information: 1. Wavelength and Refractive Index –The magnitude of the wave vector conveys the spatial period (i.e. wavelength) of the wave inside the material. When the frequency is known, the magnitude conveys the material’s refractive index n (more to be said later). 22 n k 0 free space wavelength 0 2. Direction –The direction of the wave is perpendicular to the wave fronts (more to be said later). ˆ kkakbkcabcˆˆ Slide 4 2 8/27/2020 The Dispersion Relation The dispersion relation for a material relates the wave vector to frequency . Essentially, it sets a rule for the values of as a function of direction and frequency. For an ordinary linear, homogeneous and isotropic (LHI) material, the dispersion relation is: 2 222n kkkabc c0 222 kkkabc 2 This can also be written as: 2 kk000 nc0 Slide 5 How to Derive the Dispersion Relation (1 of 2) The wave equation in a linear homogeneous anisotropic material is: 2 Assume no magnetic response i. e. 1 . Ek 00 r E 0 The solution to this equation is still a plane wave, but the allowed values for (modes) are more complicated. jk r ˆ E Ee00 E Eaabcˆˆ Eb Ec Substituting this solution into the wave equation leads to the following relation: 2 kk E000r0 kE k E 0 This equation has the form: abcˆˆ ˆ 0 Each (•••) term has the form: EEEabc 0 Each vector component must be set to zero independently.
    [Show full text]
  • 7 Plasmonics
    7 Plasmonics Highlights of this chapter: In this chapter we introduce the concept of surface plasmon polaritons (SPP). We discuss various types of SPP and explain excitation methods. Finally, di®erent recent research topics and applications related to SPP are introduced. 7.1 Introduction Long before scientists have started to investigate the optical properties of metal nanostructures, they have been used by artists to generate brilliant colors in glass artefacts and artwork, where the inclusion of gold nanoparticles of di®erent size into the glass creates a multitude of colors. Famous examples are the Lycurgus cup (Roman empire, 4th century AD), which has a green color when observing in reflecting light, while it shines in red in transmitting light conditions, and church window glasses. Figure 172: Left: Lycurgus cup, right: color windows made by Marc Chagall, St. Stephans Church in Mainz Today, the electromagnetic properties of metal{dielectric interfaces undergo a steadily increasing interest in science, dating back in the works of Gustav Mie (1908) and Rufus Ritchie (1957) on small metal particles and flat surfaces. This is further moti- vated by the development of improved nano-fabrication techniques, such as electron beam lithographie or ion beam milling, and by modern characterization techniques, such as near ¯eld microscopy. Todays applications of surface plasmonics include the utilization of metal nanostructures used as nano-antennas for optical probes in biology and chemistry, the implementation of sub-wavelength waveguides, or the development of e±cient solar cells. 208 7.2 Electro-magnetics in metals and on metal surfaces 7.2.1 Basics The interaction of metals with electro-magnetic ¯elds can be completely described within the frame of classical Maxwell equations: r ¢ D = ½ (316) r ¢ B = 0 (317) r £ E = ¡@B=@t (318) r £ H = J + @D=@t; (319) which connects the macroscopic ¯elds (dielectric displacement D, electric ¯eld E, magnetic ¯eld H and magnetic induction B) with an external charge density ½ and current density J.
    [Show full text]
  • Electromagnetism - Lecture 13
    Electromagnetism - Lecture 13 Waves in Insulators Refractive Index & Wave Impedance • Dispersion • Absorption • Models of Dispersion & Absorption • The Ionosphere • Example of Water • 1 Maxwell's Equations in Insulators Maxwell's equations are modified by r and µr - Either put r in front of 0 and µr in front of µ0 - Or remember D = r0E and B = µrµ0H Solutions are wave equations: @2E µ @2E 2E = µ µ = r r r r 0 r 0 @t2 c2 @t2 The effect of r and µr is to change the wave velocity: 1 c v = = pr0µrµ0 prµr 2 Refractive Index & Wave Impedance For non-magnetic materials with µr = 1: c c v = = n = pr pr n The refractive index n is usually slightly larger than 1 Electromagnetic waves travel slower in dielectrics The wave impedance is the ratio of the field amplitudes: Z = E=H in units of Ω = V=A In vacuo the impedance is a constant: Z0 = µ0c = 377Ω In an insulator the impedance is: µrµ0c µr Z = µrµ0v = = Z0 prµr r r For non-magnetic materials with µr = 1: Z = Z0=n 3 Notes: Diagrams: 4 Energy Propagation in Insulators The Poynting vector N = E H measures the energy flux × Energy flux is energy flow per unit time through surface normal to direction of propagation of wave: @U −2 @t = A N:dS Units of N are Wm In vacuo the amplitudeR of the Poynting vector is: 1 1 E2 N = E2 0 = 0 0 2 0 µ 2 Z r 0 0 In an insulator this becomes: 1 E2 N = N r = 0 0 µ 2 Z r r The energy flux is proportional to the square of the amplitude, and inversely proportional to the wave impedance 5 Dispersion Dispersion occurs because the dielectric constant r and refractive index
    [Show full text]
  • Chapter 5 Electromagnetic Waves in Plasmas
    Chapter 5 Electromagnetic Waves in Plasmas 5.1 General Treatment of Linear Waves in Anisotropic Medium Start with general approach to waves in a linear Medium: Maxwell: 1 ∂E ∂B � ∧ B = µ j + ; � ∧ E = − (5.1) o c2 ∂t ∂t we keep all the medium’s response explicit in j. Plasma is (infinite and) uniform so we Fourier analyze in space and time. That is we seek a solution in which all variables go like exp i(k.x − ωt) [real part of] (5.2) It is really the linearised equations which we treat this way; if there is some equilibrium field OK but the equations above mean implicitly the perturbations B, E, j, etc. Fourier analyzed: −iω ik ∧ B = µ j + E ; ik ∧ E = iωB (5.3) o c2 Eliminate B by taking k∧ second eq. and ω× 1st iω2 ik ∧ (k ∧ E) = ωµ j − E (5.4) o c2 So ω2 k ∧ (k ∧ E) + E + iωµ j = 0 (5.5) c2 o Now, in order to get further we must have some relationship between j and E(k, ω). This will have to come from solving the plasma equations but for now we can just write the most general linear relationship j and E as j = σ.E (5.6) 96 σ is the ‘conductivity tensor’. Think of this equation as a matrix e.g.: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ jx σxx σxy ... Ex ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ jy ⎠ = ⎝ ... ... ... ⎠ ⎝ Ey ⎠ (5.7) jz ... ... σzz Ez This is a general form of Ohm’s Law. Of course if the plasma (medium) is isotropic (same in all directions) all off­diagonal σ�s are zero and one gets j = σE.
    [Show full text]
  • Dispersion Relations, Linearization and Stability
    Dispersion relations, stability and linearization 1 Dispersion relations Suppose that u(x; t) is a function with domain {−∞ < x < 1; t > 0g, and it satisfies a linear, constant coefficient partial differential equation such as the usual wave or diffusion equation. It happens that these type of equations have special solutions of the form u(x; t) = exp(ikx − i!t); (1) or equivalently, u(x; t) = exp(σt + ikx): (2) We typically look for solutions of the first kind (1) when wave-like behavior which oscillates in time is expected, whereas (2) is used to investigate growth or decay in time. Plugging either (1) or (2) into the equation yields an algebraic relationship of the form ! = !(k) or σ = σ(k), called the dispersion relation. It characterizes the dynamics of spatially oscillating modes of the form exp(ikx). 2 Here are a couple standard examples. For the wave equation utt = c uxx, we plug in a wave- like solution (1) to get −!2 exp(ikx−i!t) = −c2k2 exp(ikx−i!t), or !(k) = ±ck. This means there are traveling wave solutions of the form u = exp(ik(x ± ct)), which we might have guessed from the d’Alembert formula. 2 For the diffusion equation ut = Duxx, we use (2) to get σ(k) = −Dk . Since this is negative for all k, it is expected that solutions which are superpositions of (2) also decay. This is consistent with the fundamental solution representation for the diffusion equation. 1.1 Stability For dispersion relations of the form σ = σ(k) stemming from (2), the sign of the real part of σ indicates whether the solution will grow or decay in time.
    [Show full text]
  • "Wave Optics" Lecture 21 (PDF)
    Lecture 21 Reminder/Introduction to Wave Optics Program: 1. Maxwell’s Equations. 2. Magnetic induction and electric displacement. 3. Origins of the electric permittivity and magnetic permeability. 4. Wave equations and optical constants. 5. The origins and frequency dependency of the dielectric constant. Questions you should be able to answer by the end of today’s lecture: 1. What are the equations determining propagation of the waves in the media? 2. What are the relationships between the electric field and electric displacement and magnetic field and magnetic induction? 3. What is the nature of the polarization? How can we gain intuition about the dielectric constant for a material? 4. How do Maxwell’s equations lead to the wave nature of light? 5. What is refractive index and how it relates to the dielectric constant? How does it relate to the speed of light inside the material? 6. What is the difference between the phase and group velocity? 1 Response of materials to electromagnetic waves – propagation of light in solids. We classified materials with respect to their conductivity and related the observed differences to the existence of band gaps in the electron energy eigenvalues. The optical properties are determined to a large extent by the electrons and their ability to interact with the electromagnetic field, thus the conductivity classification of materials, which was a result of the band structure and band filling, is also useful for classifying the optical response. The most widely used metric for quantifying the optical behavior of materials is the index of refraction, which captures the extent to which the electromagnetic wave interacts with the material.
    [Show full text]
  • Wave Propagation in Metamaterial Using Multiscale Resonators by Creating Local Anisotropy Raiz U
    University of South Carolina Scholar Commons Faculty Publications Mechanical Engineering, Department of 2013 Wave Propagation in Metamaterial using Multiscale Resonators by Creating Local Anisotropy Raiz U. Ahmed University of South Carolina - Columbia, [email protected] Sourav Banerjee University of South Carolina, United States, [email protected] Follow this and additional works at: https://scholarcommons.sc.edu/emec_facpub Part of the Acoustics, Dynamics, and Controls Commons, and the Applied Mechanics Commons Publication Info Published in International Journal of Modern Engineering, Volume 13, Issue 2, 2013, pages 51-59. ©International Journal of Modern Engineering 2013, International Association of Journals and Conferences. Ahmed, R. & Banerjee, S. (2013). Wave Propagation in metamaterial using Multiscale Resonators by Creating Local Anisotropy. International Journal of Modern Engineering, 13 (2),51-59. http://ijme.us/issues/spring2013/abstracts/ Z__IJME%20spring%202013%20v13%20n2%20(paper%206).pdf This Article is brought to you by the Mechanical Engineering, Department of at Scholar Commons. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. WAVE PROPAGATION IN METAMATERIAL USING MULTISCALE RESONATORS BY CREATING LOCAL ANISOTROPY ——————————————————————————————————————————————–———— Riaz Ahmed, University of South Carolina; Sourav Banerjee, University of South Carolina Abstract Introduction Directional guiding, passing or stopping of elastic waves Sonic and ultrasonic wave propagation through periodic through engineered materials have many applications to the structures has been the subject of interest for many years. engineering fields. Recently, such engineered composite Recently, this topic received greater attention in solid-state materials received great attention by the broader research physics and acoustics.
    [Show full text]
  • Phys 446: Solid State Physics / Optical Properties Lattice Vibrations: Thermal, Acoustic, and Optical Properties
    Phys 446: Solid State Physics / Optical Properties Lattice vibrations: Thermal, acoustic, and optical properties Fall 2015 Lecture 4 Andrei Sirenko, NJIT 1 Solid State Physics Lecture 4 Last weeks: (Ch. 3) • Diffraction from crystals • Scattering factors and selection rules for diffraction Today: • Lattice vibrations: Thermal, acoustic, and optical properties This Week: • Start with crystal lattice vibrations. • Elastic constants. Elastic waves. • Simple model of lattice vibrations – linear atomic chain 2 • HW1 and HW2 discussion 1 Material to be included in the 1st QZ • Crystalline structures. Diamond structure. Packing ratio 7 crystal systems and 14 Bravais lattices • Crystallographic directions n and Miller indices dhkl 1 2 h2 k 2 l 2 2 2 2 a b c • Definition of reciprocal lattice vectors: • What is Brillouin zone • Bragg formula: 2d·sinθ = mλ ; k = G 3 •Factors affecting the diffraction amplitude: Atomic scattering factor (form factor): f n(r)eikrl d 3r reflects distribution of electronic cloud. a r0 sinΔk r In case of spherical distribution f 4r 2n(r) dr a 0 Δk r •Structure factor 2i(hu j kv j lw j ) F faje j where the summation is over all atoms in unit cell •Be able to obtain scattering wave vector or frequency from geometry and data for incident beam (x-rays, neutrons or light) 4 2 Material to be included in the 2nd QZ TBD Elastic stiffness and compliance. Strain and stress: definitions and relation between them in a linear regime (Hooke's law): ij Cijkl kl ij Sijkl kl kl kl 2 2 C u C u eff •Elastic wave equation: eff x sound velocity v t 2 x2 5 • Lattice vibrations: acoustic and optical branches In three-dimensional lattice with s atoms per unit cell there are 3s phonon branches: 3 acoustic, 3s - 3 optical • Phonon - the quantum of lattice vibration.
    [Show full text]
  • Matter Waves
    Solved Problems on Matter Waves Charles Asman, Adam Monahan and Malcolm McMillan Department of Physics and Astronomy University of British Columbia, Vancouver, British Columbia, Canada Fall 1999; revised 2011 by Malcolm McMillan Given here are solutions to 6 problems on matter waves. The solutions were used as a learning-tool for students in the introductory undergraduate course Physics 200 Relativity and Quanta given by Malcolm McMillan at UBC during the 1998 and 1999 Winter Sessions. The solutions were prepared in collaboration with Charles Asman and Adam Monaham who were graduate students in the Department of Physics at the time. The problems are from Chapter 4 Matter Waves of the course text Modern Physics by Raymond A. Serway, Clement J. Moses and Curt A. Moyer, Saunders College Publishing, 2nd ed., (1997). Planck's Constant and the Speed of Light. When solving numerical problems in Quantum Mechanics is useful to note that the product of Planck's constant h = 6:6261 × 10−34 J s (1) and the speed of light c = 2:9979 × 108 m s−1 (2) is hc = 1239:8 eV nm = 1239:8 keV pm = 1239:8 MeV fm (3) where eV = 1:6022 × 10−19 J (4) Also, ~c = 197:32 eV nm = 197:32 keV pm = 197:32 MeV fm (5) where ~ = h=2π. De Broglie Matter Waves Problem 4.5, page 181 To observe small objects, one measures the diffraction of particles whose de Broglie wavelength is approxi- mately equal to the object's size. • Determine the kinetic energy (in electron volts) required for electrons to resolve a large organic molecule of size 10 nm, an atomic features of size 100 pm and a nucleus of size 10 fm.
    [Show full text]
  • Reinterpretation of Matter-Wave Interference Experiments Based on the Local-Ether Wave Equation
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Reinterpretation of Matter-Wave Interference Experiments Based on the Local-Ether Wave Equation Ching-Chuan Su Department of Electrical Engineering National Tsinghua University Hsinchu, Taiwan Abstract – Based on the local-ether wave equation for free particle, the dispersion of mat- ter wave is examined. From the dispersion relation, the angular frequency and wavelength of matter wave are derived. These formulas look like the postulates of de Broglie in conjunc- tion with the Lorentz mass-variation law. However, the fundamental difference is that for terrestrial particles their speeds are referred specifically to a geocentric inertial frame and hence incorporate the speed due to earth’s rotation. Thus the local-ether model predicts an east-west directional anisotropy both in mass and wavelength. Meanwhile, in spite of the restriction on reference frame, the local-ether model can account for the matter-wave interference experiments of the Bragg reflection and the Sagnac effect. For electron wave, the effects of earth’s rotation are negligible and the derived Bragg angle is actually in accord with the Davisson-Germer experiment, as examined within the present precision. On the other hand, the local-ether model leads to a directional anisotropy in the Bragg angle in neutron diffraction. The predicted anisotropy due to earth’s rotation then provide a means to test the local-ether wave equation. 1. Introduction The wave nature of particle has been initiated by de Broglie in 1924 by postulating that a particle is associated with a matter wave of which the angular frequency and the wavelength are related to the energy and the momentum of the particle, respectively [1].
    [Show full text]