Roadmap on Transformation Optics 5 Martin Mccall 1,*, John B Pendry 1, Vincenzo Galdi 2, Yun Lai 3, S
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
Electrostatics Vs Magnetostatics Electrostatics Magnetostatics
Electrostatics vs Magnetostatics Electrostatics Magnetostatics Stationary charges ⇒ Constant Electric Field Steady currents ⇒ Constant Magnetic Field Coulomb’s Law Biot-Savart’s Law 1 ̂ ̂ 4 4 (Inverse Square Law) (Inverse Square Law) Electric field is the negative gradient of the Magnetic field is the curl of magnetic vector electric scalar potential. potential. 1 ′ ′ ′ ′ 4 |′| 4 |′| Electric Scalar Potential Magnetic Vector Potential Three Poisson’s equations for solving Poisson’s equation for solving electric scalar magnetic vector potential potential. Discrete 2 Physical Dipole ′′′ Continuous Magnetic Dipole Moment Electric Dipole Moment 1 1 1 3 ∙̂̂ 3 ∙̂̂ 4 4 Electric field cause by an electric dipole Magnetic field cause by a magnetic dipole Torque on an electric dipole Torque on a magnetic dipole ∙ ∙ Electric force on an electric dipole Magnetic force on a magnetic dipole ∙ ∙ Electric Potential Energy Magnetic Potential Energy of an electric dipole of a magnetic dipole Electric Dipole Moment per unit volume Magnetic Dipole Moment per unit volume (Polarisation) (Magnetisation) ∙ Volume Bound Charge Density Volume Bound Current Density ∙ Surface Bound Charge Density Surface Bound Current Density Volume Charge Density Volume Current Density Net , Free , Bound Net , Free , Bound Volume Charge Volume Current Net , Free , Bound Net ,Free , Bound 1 = Electric field = Magnetic field = Electric Displacement = Auxiliary -
Transformation-Optics-Based Design of a Metamaterial Radome For
IEEE JOURNAL ON MULTISCALE AND MULTIPHYSICS COMPUTATIONAL TECHNIQUES, VOL. 2, 2017 159 Transformation-Optics-Based Design of a Metamaterial Radome for Extending the Scanning Angle of a Phased-Array Antenna Massimo Moccia, Giuseppe Castaldi, Giuliana D’Alterio, Maurizio Feo, Roberto Vitiello, and Vincenzo Galdi, Fellow, IEEE Abstract—We apply the transformation-optics approach to the the initial focus on EM, interest has rapidly spread to other design of a metamaterial radome that can extend the scanning an- disciplines [4], and multiphysics applications are becoming in- gle of a phased-array antenna. For moderate enhancement of the creasingly relevant [5]–[7]. scanning angle, via suitable parameterization and optimization of the coordinate transformation, we obtain a design that admits a Metamaterial synthesis has several traits in common technologically viable, robust, and potentially broadband imple- with inverse-scattering problems [8], and likewise poses mentation in terms of thin-metallic-plate inclusions. Our results, some formidable computational challenges. Within the validated via finite-element-based numerical simulations, indicate emerging framework of “metamaterial-by-design” [9], the an alternative route to the design of metamaterial radomes that “transformation-optics” (TO) approach [10], [11] stands out does not require negative-valued and/or extreme constitutive pa- rameters. as a systematic strategy to analytically derive the idealized material “blueprints” necessary to implement a desired field- Index Terms—Metamaterials, -
Review of Electrostatics and Magenetostatics
Review of electrostatics and magenetostatics January 12, 2016 1 Electrostatics 1.1 Coulomb’s law and the electric field Starting from Coulomb’s law for the force produced by a charge Q at the origin on a charge q at x, qQ F (x) = 2 x^ 4π0 jxj where x^ is a unit vector pointing from Q toward q. We may generalize this to let the source charge Q be at an arbitrary postion x0 by writing the distance between the charges as jx − x0j and the unit vector from Qto q as x − x0 jx − x0j Then Coulomb’s law becomes qQ x − x0 x − x0 F (x) = 2 0 4π0 jx − xij jx − x j Define the electric field as the force per unit charge at any given position, F (x) E (x) ≡ q Q x − x0 = 3 4π0 jx − x0j We think of the electric field as existing at each point in space, so that any charge q placed at x experiences a force qE (x). Since Coulomb’s law is linear in the charges, the electric field for multiple charges is just the sum of the fields from each, n X qi x − xi E (x) = 4π 3 i=1 0 jx − xij Knowing the electric field is equivalent to knowing Coulomb’s law. To formulate the equivalent of Coulomb’s law for a continuous distribution of charge, we introduce the charge density, ρ (x). We can define this as the total charge per unit volume for a volume centered at the position x, in the limit as the volume becomes “small”. -
Electromagnetism What Is the Effect of the Number of Windings of Wire on the Strength of an Electromagnet?
TEACHER’S GUIDE Electromagnetism What is the effect of the number of windings of wire on the strength of an electromagnet? GRADES 6–8 Physical Science INQUIRY-BASED Science Electromagnetism Physical Grade Level/ 6–8/Physical Science Content Lesson Summary In this lesson students learn how to make an electromagnet out of a battery, nail, and wire. The students explore and then explain how the number of turns of wire affects the strength of an electromagnet. Estimated Time 2, 45-minute class periods Materials D cell batteries, common nails (20D), speaker wire (18 gauge), compass, package of wire brad nails (1.0 mm x 12.7 mm or similar size), Investigation Plan, journal Secondary How Stuff Works: How Electromagnets Work Resources Jefferson Lab: What is an electromagnet? YouTube: Electromagnet - Explained YouTube: Electromagnets - How can electricity create a magnet? NGSS Connection MS-PS2-3 Ask questions about data to determine the factors that affect the strength of electric and magnetic forces. Learning Objectives • Students will frame a hypothesis to predict the strength of an electromagnet due to changes in the number of windings. • Students will collect and analyze data to determine how the number of windings affects the strength of an electromagnet. What is the effect of the number of windings of wire on the strength of an electromagnet? Electromagnetism is one of the four fundamental forces of the universe that we rely on in many ways throughout our day. Most home appliances contain electromagnets that power motors. Particle accelerators, like CERN’s Large Hadron Collider, use electromagnets to control the speed and direction of these speedy particles. -
Transformation Optics for Thermoelectric Flow
J. Phys.: Energy 1 (2019) 025002 https://doi.org/10.1088/2515-7655/ab00bb PAPER Transformation optics for thermoelectric flow OPEN ACCESS Wencong Shi, Troy Stedman and Lilia M Woods1 RECEIVED 8 November 2018 Department of Physics, University of South Florida, Tampa, FL 33620, United States of America 1 Author to whom any correspondence should be addressed. REVISED 17 January 2019 E-mail: [email protected] ACCEPTED FOR PUBLICATION Keywords: thermoelectricity, thermodynamics, metamaterials 22 January 2019 PUBLISHED 17 April 2019 Abstract Original content from this Transformation optics (TO) is a powerful technique for manipulating diffusive transport, such as heat work may be used under fl the terms of the Creative and electricity. While most studies have focused on individual heat and electrical ows, in many Commons Attribution 3.0 situations thermoelectric effects captured via the Seebeck coefficient may need to be considered. Here licence. fi Any further distribution of we apply a uni ed description of TO to thermoelectricity within the framework of thermodynamics this work must maintain and demonstrate that thermoelectric flow can be cloaked, diffused, rotated, or concentrated. attribution to the author(s) and the title of Metamaterial composites using bilayer components with specified transport properties are presented the work, journal citation and DOI. as a means of realizing these effects in practice. The proposed thermoelectric cloak, diffuser, rotator, and concentrator are independent of the particular boundary conditions and can also operate in decoupled electric or heat modes. 1. Introduction Unprecedented opportunities to manipulate electromagnetic fields and various types of transport have been discovered recently by utilizing metamaterials (MMs) capable of achieving cloaking, rotating, and concentrating effects [1–4]. -
Controlling the Wave Propagation Through the Medium Designed by Linear Coordinate Transformation
Home Search Collections Journals About Contact us My IOPscience Controlling the wave propagation through the medium designed by linear coordinate transformation This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Eur. J. Phys. 36 015006 (http://iopscience.iop.org/0143-0807/36/1/015006) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 128.42.82.187 This content was downloaded on 04/07/2016 at 02:47 Please note that terms and conditions apply. European Journal of Physics Eur. J. Phys. 36 (2015) 015006 (11pp) doi:10.1088/0143-0807/36/1/015006 Controlling the wave propagation through the medium designed by linear coordinate transformation Yicheng Wu, Chengdong He, Yuzhuo Wang, Xuan Liu and Jing Zhou Applied Optics Beijing Area Major Laboratory, Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China E-mail: [email protected] Received 26 June 2014, revised 9 September 2014 Accepted for publication 16 September 2014 Published 6 November 2014 Abstract Based on the principle of transformation optics, we propose to control the wave propagating direction through the homogenous anisotropic medium designed by linear coordinate transformation. The material parameters of the medium are derived from the linear coordinate transformation applied. Keeping the space area unchanged during the linear transformation, the polarization-dependent wave control through a non-magnetic homogeneous medium can be realized. Beam benders, polarization splitter, and object illu- sion devices are designed, which have application prospects in micro-optics and nano-optics. -
Transformation Magneto-Statics and Illusions for Magnets
OPEN Transformation magneto-statics and SUBJECT AREAS: illusions for magnets ELECTRONIC DEVICES Fei Sun1,2 & Sailing He1,2 TRANSFORMATION OPTICS APPLIED PHYSICS 1 Centre for Optical and Electromagnetic Research, Zhejiang Provincial Key Laboratory for Sensing Technologies, JORCEP, East Building #5, Zijingang Campus, Zhejiang University, Hangzhou 310058, China, 2Department of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology (KTH), S-100 44 Stockholm, Sweden. Received 26 June 2014 Based on the form-invariant of Maxwell’s equations under coordinate transformations, we extend the theory Accepted of transformation optics to transformation magneto-statics, which can design magnets through coordinate 28 August 2014 transformations. Some novel DC magnetic field illusions created by magnets (e.g. rescaling magnets, Published cancelling magnets and overlapping magnets) are designed and verified by numerical simulations. Our 13 October 2014 research will open a new door to designing magnets and controlling DC magnetic fields. ransformation optics (TO), which has been utilized to control the path of electromagnetic waves1–7, the Correspondence and conduction of current8,9, and the distribution of DC electric or magnetic field10–20 in an unprecedented way, requests for materials T has become a very popular research topic in recent years. Based on the form-invariant of Maxwell’s equation should be addressed to under coordinate transformations, special media (known as transformed media) with pre-designed functionality have been designed by using coordinate transformations1–4. TO can also be used for designing novel plasmonic S.H. ([email protected]) nanostructures with broadband response and super-focusing ability21–23. By analogy to Maxwell’s equations, the form-invariant of governing equations of other fields (e.g. -
Lecture 8: Magnets and Magnetism Magnets
Lecture 8: Magnets and Magnetism Magnets •Materials that attract other metals •Three classes: natural, artificial and electromagnets •Permanent or Temporary •CRITICAL to electric systems: – Generation of electricity – Operation of motors – Operation of relays Magnets •Laws of magnetic attraction and repulsion –Like magnetic poles repel each other –Unlike magnetic poles attract each other –Closer together, greater the force Magnetic Fields and Forces •Magnetic lines of force – Lines indicating magnetic field – Direction from N to S – Density indicates strength •Magnetic field is region where force exists Magnetic Theories Molecular theory of magnetism Magnets can be split into two magnets Magnetic Theories Molecular theory of magnetism Split down to molecular level When unmagnetized, randomness, fields cancel When magnetized, order, fields combine Magnetic Theories Electron theory of magnetism •Electrons spin as they orbit (similar to earth) •Spin produces magnetic field •Magnetic direction depends on direction of rotation •Non-magnets → equal number of electrons spinning in opposite direction •Magnets → more spin one way than other Electromagnetism •Movement of electric charge induces magnetic field •Strength of magnetic field increases as current increases and vice versa Right Hand Rule (Conductor) •Determines direction of magnetic field •Imagine grasping conductor with right hand •Thumb in direction of current flow (not electron flow) •Fingers curl in the direction of magnetic field DO NOT USE LEFT HAND RULE IN BOOK Example Draw magnetic field lines around conduction path E (V) R Another Example •Draw magnetic field lines around conductors Conductor Conductor current into page current out of page Conductor coils •Single conductor not very useful •Multiple winds of a conductor required for most applications, – e.g. -
Design Method for Transformation Optics Based on Laplace's Equation
Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries Zheng Chang1, Xiaoming Zhou1, Jin Hu2 and Gengkai Hu 1* 1School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, P. R. China 2School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, P. R. China *Corresponding author: [email protected] Abstract: Recently, there are emerging demands for isotropic material parameters, arising from the broadband requirement of the functional devices. Since inverse Laplace’s equation with sliding boundary condition will determine a quasi-conformal mapping, and a quasi-conformal mapping will minimize the transformation material anisotropy, so in this work, the inverse Laplace’s equation with sliding boundary condition is proposed for quasi-isotropic transformation material design. Examples of quasi-isotropic arbitrary carpet cloak and waveguide with arbitrary cross sections are provided to validate the proposed method. The proposed method is very simple compared with other quasi-conformal methods based on grid generation tools. 1. Introduction The form invariance of Maxwell’s equations under coordinate transformations constructs the equivalence between the geometrical space and material space, the developed transformation optics theory [1,2] is opening a new field that may cover many aspects of physics and wave phenomenon. Transformation optics (TO) makes it possible to investigate celestial motion and light/matter behavior in laboratory environment by using equivalent materials [3]. Illusion optics [4] is also proposed based on transformation optics, which can design a device looking differently. The attractive application of transformation optics is the invisibility cloak, which covers objects without being detected [5]. -
We Continue to Compare the Electrostatic and Magnetostatic Fields. the Electrostatic Field Is Conservative
We continue to compare the electrostatic and magnetostatic fields. The electrostatic field is conservative: This allows us to define the potential V: dl because a b is independent of the path. If a vector field has no curl (i.e., is conservative), it must be something's gradient. Gravity is conservative. Therefore you do see water flowing in such a loop without a pump in the physical world. For the magnetic field, If a vector field has no divergence (i.e., is solenoidal), it must be something's curl. In other words, the curl of a vector field has zero divergence. Let’s use another physical context to help you understand this math: Ampère’s law J ds 0 Kirchhoff's current law (KCL) S Since , we can define a vector field A such that Notice that for a given B, A is not unique. For example, if then , because Similarly, for the electrostatic field, the scalar potential V is not unique: If then You have the freedom to choose the reference (Ampère’s law) Going through the math, you will get Here is what means: Just notation. Notice that is a vector. Still remember what means for a scalar field? From a previous lecture: Recall that the choice for A is not unique. It turns out that we can always choose A such that (Ampère’s law) The choice for A is not unique. We choose A such that Here is what means: Notice that is a vector. Thus this is actually three equations: Recall the definition of for a scalar field from a previous lecture: Poisson’s equation for the magnetic field is actually three equations: Compare Poisson’s equation for the magnetic field with that for the electrostatic field: Given J, you can solve A, from which you get B by Given , you can solve V, from which you get E by Exams (Test 2 & Final) problems will not involve the vector potential. -
A Unified Approach to Nonlinear Transformation Materials
www.nature.com/scientificreports OPEN A Unifed Approach to Nonlinear Transformation Materials Sophia R. Sklan & Baowen Li The advances in geometric approaches to optical devices due to transformation optics has led to the Received: 9 October 2017 development of cloaks, concentrators, and other devices. It has also been shown that transformation Accepted: 9 January 2018 optics can be used to gravitational felds from general relativity. However, the technique is currently Published: xx xx xxxx constrained to linear devices, as a consistent approach to nonlinearity (including both the case of a nonlinear background medium and a nonlinear transformation) remains an open question. Here we show that nonlinearity can be incorporated into transformation optics in a consistent way. We use this to illustrate a number of novel efects, including cloaking an optical soliton, modeling nonlinear solutions to Einstein’s feld equations, controlling transport in a Debye solid, and developing a set of constitutive to relations for relativistic cloaks in arbitrary nonlinear backgrounds. Transformation optics1–9, which uses geometric coordinate transformations derive the materials requirements of arbitrary devices, is a powerful technique. Essentially, for any geometry there corresponds a material with identi- cal transport. With the correct geometry, it is possible to construct optical cloaks10–12 and concentrators13 as well as analogues of these devices for other waves14–18 and even for difusion19–26. While many interpretations and for- malisms of transformation optics exist, such as Jacobian transformations4, scattering matrices27–31, and conformal mappings3, one of the most theoretically powerful interpretations comes from the metric formalism32. All of these approaches agree that materials defne an efective geometry, however the metric formalism is important since it allows us to further interpret the geometry.