DOCSLIB.ORG

Optics, Lecture 9

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Optics, Lecture 9 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang Outline: A. Electromagnetism B. Frequency Domain (Fourier transform) C. EM waves in Cartesian coordinates D. Energy Flow and Poynting Vector E. Connection to geometrical optics F. Eikonal Equations: Path of Light in an Inhomogeneous Medium A. Electromagnetism and Maxwell Equations, Differential Forms: - In real space, time-dependent fields: D (displacement field) D dS dS + V A Coulomb (Coulomb’s Law, electric field) (1) dS there are no B magnetic charges V A (2) (Gauss’ Law, magnetic field) 1 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang E dl B(t) up or down A C − 푡 (3) (Faraday’s Law) H H dl dl D I current capacitor A CA C 푡 (4) (Ampere-Maxwell’s Law) Note: J, q are sources of EM radiation and E, D, H, B are induced fields. B. From time domain to frequency domain (Fourier Transform): Continuous wave laser light field understudy are often mono-chromatic. These problems are mapped in the Maxwell equations by expanding complex time signals to a series of time harmonic components (often referred to as “single” wavelength light): ⃗ ( ) ∞ ⃗⃗ ( ) ( ) e.g. 푟 , 푡 ∫−∞ 푬 푟 , 푒푥푝 −푖푡 푑 (5) Advantage: 휕 ∞ 휕 ∞ ⃗ (푟 , 푡) ∫ 푬⃗⃗ (푟 , ) 푒푥푝(−푖푡)푑 ∫ [−풊흎푬⃗⃗ (⃗풓 , 흎)]푒푥푝(−푖푡)푑 (6) 휕푡 −∞ 휕푡 −∞ 2 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang 휕 So, we can replace all time derivatives by −푖 in frequency domain: 휕푡 (Faraday’s Law:) (7) (Ampere-Maxwell’s Law:) (8) The other two equations remain unaltered. There are total of 12 unknowns (E, H, D, B) but so far we only obtained 8 equations from the Maxwell equations (2 vector form x3 + 2 scalar forms) so more information needed to understand the complete wave behavior! Generally we may start to construct the response of a material by applying a excitation field E or H in vacuum. Therefore it is more typical to consider the E, H field as input and D, B fields as output. In common optical materials, we may enjoy the following simplification of local (i.e. independent of neighbors) and linear relationship: 푫⃗⃗ () 휀()휀0푬⃗⃗ () (9) 푩⃗⃗ () 휇()휇0푯⃗⃗⃗ () (10) The so called (electric) permittivity 휀() and (magnetic) permeability 휇()are unitless parameters that depend on the frequency of the input field. In the case of anisotropic medium, both 휀() and 휇()become 3x3 dimension tensor. Now we have 6 more equations from material response, we can include them together with Maxwell equations to obtain a complete solution of optical fields with proper boundary condition. C. Maxwell’s Equations in Cartesian Coordinates: To solve Maxwell equations in Cartesian coordinates, we need to practice on the vector operators accordingly. Most unfamiliar one is probably the curl of a vector . In Cartesian coordinates it is often written as a matrix determinant: 3 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang 풙̂ 풚̂ 풛̂ ⃗⃗ 휕 휕 휕 훁 푨 | 휕푥 휕푦 휕푧 | (11) 풙 풚 풛 In this fashion, we may write the Faraday’s law in frequency domain, , with 3 components in Cartesian coordinates: 휕퐸 휕퐸 ( 푧 − 푦) 푖 (12) 휕푦 휕푧 푥 휕퐸 휕퐸 ( 푧 − 푥) 푖 (13) 휕푥 휕푧 푦 휕퐸 휕퐸 ( 푦 − 푥) 푖 (14) 휕푥 휕푦 푧 Likewise, we arrive at the rest of Maxwell’s equations: 휕퐻 휕퐻 ( 푧 − 푦) −푖 (15) 휕푦 휕푧 푥 휕퐻 휕퐻 ( 푧 − 푥) −푖 (16) 휕푥 휕푧 푦 휕퐻 휕퐻 ( 푦 − 푥) −푖 (17) 휕푥 휕푦 푧 together with 휕 휕 휕 (18) 휕푥 푥 휕푦 푦 휕푧 푧 And 휕 휕 휕 (19) 휕푥 푥 휕푦 푦 휕푧 푧 - Example: Plane EM wave in 1-D homogeneous medium (e.g. an expanded 흏 흏 laser beam in +z direction, ퟎ, ퟎ) 흏풙 흏풚 We can now further simplify from the above equations in Cartesian coordinates: 휕퐸 휕퐸 ( 푧 − 푦) 푖 (20) 휕푦 휕푧 푥 휕퐸 휕퐸 ( 푧 − 푥) 푖 (21) 휕푥 휕푧 푦 And 휕퐻 휕퐻 ( 푧 − 푦) −푖 (22) 휕푦 휕푧 푥 휕퐻 휕퐻 ( 푧 − 푥) −푖 (23) 휕푥 휕푧 푦 From the above we found only 4 non-trivial equations. In the isotropic case, they can be further divided into 2 independent sub-groups (two Polarizations!): 4 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang (Ex, Hy only) 휕 푖휇휇 − (24) 0 푦 휕푧 푥 휕 and 푖휀휀 (25) 0 푧 휕푧 푦 Or (Ey, Hx only) 휕 푖휇휇 − (26) 0 푥 휕푧 푦 휕 And 푖휀휀 (27) 0 푦 휕푧 푥 Observations: - We see that wave propagation in such medium is purely transverse, i.e. only components of E, H field that are orthogonal to propagation direction (+z) survived in the wave field. 흏 - Taking the derivative again on any of these equations, we obtain wave 흏풛 equation such as: 휕2 휕 −푖휇휇 −푖휇휇 (푖휀휀 ) (28) 휕푧2 푥 0 휕푧 푦 0 0 푥 2 2 휕 2 휀휇휔 2 푥 ( 휇0휀0)휀휇푥 ( 2 ) 푥 (29) 휕푧 푐0 2 1 since the speed of light c0 in vacuum satisfy ( c0 ) 휀0휇0 Therefore the index of refraction is found as: 푛() √휀()휇() (30) D. The Poynting Vector ⃗ 2 The Poynting vector, 푺 휀0푐0 ⃗ ⃗ , is used to power per unit area in the direction of propagation. 5 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang o Justification: U = Energy density Energy passing through area A in time t: A So the energy per unit time per unit area: c t - The Irradiance (often called the Intensity) Visible light wave oscillates in 1014-1015Hz. Since we don’t have a detector that responds in such a high speed yet, for convenience we take the average power per unit area, as the irradiance. Substituting a sinusoidal light wave into the expression for the Poynting vector yields 1 〈⃗푺 〉 휀 푐2〈⃗ ⃗ 〉 휀 푐2 (31) 0 0 2 0 0 0 0 E. High Frequency Limit, connection to Geometric Optics: How can we obtain Geometric optics picture such as ray tracing from wave equations? Now let’s go back to real space and frequency domain (in an isotropic medium but with spatially varying permittivity 휀(푥, 푧), for example). 휕2 휕2 휔2 2 푥 2 푥 휀(푥, 푧) ( 2 ) 푥 (32) 휕푥 휕푧 푐0 Now we decompose the field E(r, ) into two forms: a fast oscillating component exp(ik0), k0 /푐0 and a slowly varying envelope E0(r) as illustrated in the textbox. 6 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang With this tentative solution, we can rewrite the wave equation: 휕 휕 휕 ( (푥, 푧)) exp(푖푘Φ(푥, 푧)) (푥, 푧) [ exp(푖푘Φ(푥, 푧))] (33) 휕푥 푥 휕푥 0 0 휕푥 휕 휕 휕Φ(푥,푧) ( (푥, 푧)) exp(푖푘Φ(푥, 푧)) (푥, 푧) [푖푘 ] exp(푖푘Φ(푥, 푧)) (34) 휕푥 푥 휕푥 0 0 휕푥 휕 휕 휕Φ(푥,푧) [ (푥, 푧) 푖푘 (푥, 푧) ] exp(푖푘Φ(푥, 푧)) (35) 휕푥 푥 휕푥 0 0 휕푥 휕2 휕2 휕2 휕 휕 [ (푥, 푧) 푖푘 (푥, 푧) Φ(푥, 푧) 2푖푘 (푥, 푧) Φ(푥, 푧) − 휕푥2 푥 휕푥2 0 0 휕푥2 휕푥 0 휕푥 2 휕 푘2 (푥, 푧) ( Φ(푥, 푧)) ] exp(푖푘Φ(푥, 푧)) (36) 0 휕푥 Similarly you can find the derivative along the z direction. So our wave equation becomes: 2 2 휕 휕 푘2 [휀(푥, 푧) − ( Φ(푥, 푧)) − ( Φ(푥, 푧)) ] (푥, 푧) 휕푥 휕푧 0 휕2 휕2 휕 휕 휕 휕 [ (푥, 푧) (푥, 푧)] 2푖푘 [ (푥, 푧) Φ(푥, 푧) (푥, 푧) Φ(푥, 푧)] 휕푥2 0 휕푧2 0 휕푥 0 휕푥 휕푧 0 휕푧 휕2 휕2 푖푘 (푥, 푧) [ Φ(푥, 푧) Φ(푥, 푧)] (37) 0 휕푥2 휕푧2 Furthermore, if the envelope of field varies slowly with E 1 휕 1 휕 0 ∇Φ wavelength (e.g. 0 ≪ 1, 0 ≪ 1) then only 푘 휕푥 푘 휕푧 H0 the first term is important. 2 2 3 휕Φ 휕Φ ( ) ( ) 휀(푥, 푧) 푛2(푥, 푧) (38) 2 휕푥 휕푧 1 This is the well-known Eikonal equation, Φ being the Geometrical relationship eikonal (derived from a Greek word, meaning image). of E, H, and ∇Φ 7 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang F. Path of Light in an Inhomogeneous Medium A. Example 1: 1D problems (Gradient index waveguides, Mirage Effects) The best known example of this kind is probably the Mirage effect in desert or near a seashore, and we heard of the explanation such as the refractive index increases with density (and hence decreases with temperature at a given altitude). With the picture in mind, now can we predict more accurately the ray path and image forming processes? Starting from the Eikonal equation and we assume 푛2(푥, 푧) is only a function of x, then we find: 휕Φ 2 휕Φ 2 ( ) ( ) 푛2(푥) (39) 휕푥 휕푧 Since there is the index in independent of z, we may assume the slope of phase change in z direction is linear: 휕Φ ( )= C(const) (40) 휕푧 This allows us to find 휕Φ 2 2 √푛 (푥) − (41) 휕푥 From Fermat’s principle, we can visualize that direction of rays follow the gradient of phase front: 푑푟 푛 ∇Φ (42) 푑푙 8 Lecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics –Nick Fang 푑푧 z-direction: 푛(푥) C (43) 푑푙 푑푥 2 x-direction: 푛(푥) √푛2(푥) − (44) 푑푙 Therefore, the light path (x, z) is determined by: 푑푧 C (45) 푑푥 √푛2(푥)−퐶2 Hence 푥 C 푧 − 푧0 ∫ 푑푥 (46) 푥0 √푛2(푥)−퐶2 Without loss of generality, we may assume a quadratic index profile along the x direction, such as found in gradient index optical fibers or rods: 2 2 2 푛 (푥) 푛0(1 − 훼푥 ) (47) 푥 C 푧 − 푧0 ∫ 푑푥 (48) 푥0 2 2 2 √푛 (1−훼푥 )−퐶 To find the integral explicitly we may take the following transformation of the variable x: 2 2 푛0−퐶 푥 √ 2 푠푖푛휃 (49) 푛0훼 Therefore, 휃 C 푧 − 푧0 ∫ 푑휃 (50) 휃0 푛0√훼 C 푧 푧0 (휃 − 휃0) (51) 푛0√훼 Or more commonly, 2 2 푛0훼 √푛0훼 푥√ 2 2 푠푖푛휃 푠푖푛 (휃0 2 (푧 − 푧0)) (52) 푛0−퐶 퐶 As you can see in this example, ray propagation in the gradient index waveguide follows a 2휋퐶 sinusoid pattern! The periodicity is determined by a constant .
Load more

Recommended publications
  • Descartes' Optics
    Descartes’ Optics Jeffrey K. McDonough Descartes’ work on optics spanned his entire career and represents a fascinating area of inquiry. His interest in the study of light is already on display in an intriguing study of refraction from his early notebook, known as the Cogitationes privatae, dating from 1619 to 1621 (AT X 242-3). Optics figures centrally in Descartes’ The World, or Treatise on Light, written between 1629 and 1633, as well as, of course, in his Dioptrics published in 1637. It also, however, plays important roles in the three essays published together with the Dioptrics, namely, the Discourse on Method, the Geometry, and the Meteorology, and many of Descartes’ conclusions concerning light from these earlier works persist with little substantive modification into the Principles of Philosophy published in 1644. In what follows, we will look in a brief and general way at Descartes’ understanding of light, his derivations of the two central laws of geometrical optics, and a sampling of the optical phenomena he sought to explain. We will conclude by noting a few of the many ways in which Descartes’ efforts in optics prompted – both through agreement and dissent – further developments in the history of optics. Descartes was a famously systematic philosopher and his thinking about optics is deeply enmeshed with his more general mechanistic physics and cosmology. In the sixth chapter of The Treatise on Light, he asks his readers to imagine a new world “very easy to know, but nevertheless similar to ours” consisting of an indefinite space filled everywhere with “real, perfectly solid” matter, divisible “into as many parts and shapes as we can imagine” (AT XI ix; G 21, fn 40) (AT XI 33-34; G 22-23).
    [Show full text]
  • Rays, Waves, and Scattering: Topics in Classical Mathematical Physics
    chapter1 February 28, 2017 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Chapter One Introduction Probably no mathematical structure is richer, in terms of the variety of physical situations to which it can be applied, than the equations and techniques that constitute wave theory. Eigenvalues and eigenfunctions, Hilbert spaces and abstract quantum mechanics, numerical Fourier analysis, the wave equations of Helmholtz (optics, sound, radio), Schrödinger (electrons in matter) ... variational methods, scattering theory, asymptotic evaluation of integrals (ship waves, tidal waves, radio waves around the earth, diffraction of light)—examples such as these jostle together to prove the proposition. M. V. Berry [1] There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened. Douglas Adams [155] Douglas Adams’s famous Hitchhiker trilogy consists of five books; coincidentally this book addresses the three topics of rays, waves, and scattering in five parts: (i) Rays, (ii) Waves, (iii) Classical Scattering, (iv) Semiclassical Scattering, and (v) Special Topics in Scattering Theory (followed by six appendices, some of which deal with more specialized topics). I have tried to present a coherent account of each of these topics by separating them insofar as it is possible, but in a very real sense they are inseparable. We are in effect viewing each phenomenon (e.g.
    [Show full text]
  • Studying Charged Particle Optics: an Undergraduate Course
    IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 29 (2008) 251–256 doi:10.1088/0143-0807/29/2/007 Studying charged particle optics: an undergraduate course V Ovalle1,DROtomar1,JMPereira2,NFerreira1, RRPinho3 and A C F Santos2,4 1 Instituto de Fisica, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n◦. Gragoata,´ Niteroi,´ 24210-346 Rio de Janeiro, Brazil 2 Instituto de Fisica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, Brazil 3 Departamento de F´ısica–ICE, Universidade Federal de Juiz de Fora, Campus Universitario,´ 36036-900, Juiz de Fora, MG, Brazil E-mail: [email protected] (A C F Santos) Received 23 August 2007, in final form 12 December 2007 Published 17 January 2008 Online at stacks.iop.org/EJP/29/251 Abstract This paper describes some computer-based activities to bring the study of charged particle optics to undergraduate students, to be performed as a part of a one-semester accelerator-based experimental course. The computational simulations were carried out using the commercially available SIMION program. The performance parameters, such as the focal length and P–Q curves are obtained. The three-electrode einzel lens is exemplified here as a study case. Introduction For many decades, physicists have been employing charged particle beams in order to investigate elementary processes in nuclear, atomic and particle physics using accelerators. In addition, many areas such as biology, chemistry, engineering, medicine, etc, have benefited from using beams of ions or electrons so that their phenomenological aspects can be understood. Among all its applications, electron microscopy may be one of the most important practical applications of lenses for charged particles.
    [Show full text]
  • Geodesic Conformal Transformation Optics: Manipulating Light With
    Geodesic conformal transformation optics: manipulating light with continuous refractive index profile Lin Xu1 , Tomáš Tyc 2 and Huanyang Chen1* 1 Institute of Electromagnetics and Acoustics and Department of Electronic Science, Xiamen University Xiamen 361005, China 2Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlarska 2, 61137 Brno, Czech Republic Conformal transformation optics provides a simple scheme for manipulating light rays with inhomogeneous isotropic dielectrics. However, there is usually discontinuity for refractive index profile at branch cuts of different virtual Riemann sheets, hence compromising the functionalities. To deal with that, we present a special method for conformal transformation optics based on the concept of geodesic lens. The requirement is a continuous refractive index profile of dielectrics, which shows almost perfect performance of designed devices. We demonstrate such a proposal by achieving conformal transparency and reflection. We can further achieve conformal invisible cloaks by two techniques with perfect electromagnetic conductors. The geodesic concept may also find applications in other waves that obey the Helmholtz equation in two dimensions. Introduction.-Based on covariance of Maxwell’s equations and multi-linear constitutive equations, optical property of virtual space and physical space could be connected by a coordinate mapping [1]. In 2006, Leonhardt [2] presented that a conformal coordinate mapping between two complex planes could be performed for scalar field of refractive index of dielectrics such that light rays could be manipulated freely. Coincidentally, Pendry et al [3] provided a general method for controlling electromagnetic field in space of three dimensions. These two seminal papers launched a new research field named transformation optics (TO) [4-7], which originally mainly focused on optical invisibility.
    [Show full text]
  • The Ray Optics Module User's Guide
    Ray Optics Module User’s Guide Ray Optics Module User’s Guide © 1998–2018 COMSOL Protected by patents listed on www.comsol.com/patents, and U.S. Patents 7,519,518; 7,596,474; 7,623,991; 8,457,932; 8,954,302; 9,098,106; 9,146,652; 9,323,503; 9,372,673; and 9,454,625. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/comsol-license-agreement) and may be used or copied only under the terms of the license agreement. COMSOL, the COMSOL logo, COMSOL Multiphysics, COMSOL Desktop, COMSOL Server, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/trademarks. Version: COMSOL 5.4 Contact Information Visit the Contact COMSOL page at www.comsol.com/contact to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at www.comsol.com/contact/offices for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at www.comsol.com/support/case. Other useful links include: • Support Center: www.comsol.com/support • Product Download: www.comsol.com/product-download • Product Updates: www.comsol.com/support/updates • COMSOL Blog: www.comsol.com/blogs • Discussion Forum: www.comsol.com/community • Events: www.comsol.com/events • COMSOL Video Gallery: www.comsol.com/video • Support Knowledge Base: www.comsol.com/support/knowledgebase Part number: CM024201 Contents Chapter 1: Introduction About the Ray Optics Module 8 The Ray Optics Module Physics Interface Guide .
    [Show full text]
  • Geometric Optics 1 7.1 Overview
    Contents III OPTICS ii 7 Geometric Optics 1 7.1 Overview...................................... 1 7.2 Waves in a Homogeneous Medium . 2 7.2.1 Monochromatic, Plane Waves; Dispersion Relation . ........ 2 7.2.2 WavePackets ............................... 4 7.3 Waves in an Inhomogeneous, Time-Varying Medium: The Eikonal Approxi- mationandGeometricOptics . .. .. .. 7 7.3.1 Geometric Optics for a Prototypical Wave Equation . ....... 8 7.3.2 Connection of Geometric Optics to Quantum Theory . ..... 11 7.3.3 GeometricOpticsforaGeneralWave . .. 15 7.3.4 Examples of Geometric-Optics Wave Propagation . ...... 17 7.3.5 Relation to Wave Packets; Breakdown of the Eikonal Approximation andGeometricOptics .......................... 19 7.3.6 Fermat’sPrinciple ............................ 19 7.4 ParaxialOptics .................................. 23 7.4.1 Axisymmetric, Paraxial Systems; Lenses, Mirrors, Telescope, Micro- scopeandOpticalCavity. 25 7.4.2 Converging Magnetic Lens for Charged Particle Beam . ....... 29 7.5 Catastrophe Optics — Multiple Images; Formation of Caustics and their Prop- erties........................................ 31 7.6 T2 Gravitational Lenses; Their Multiple Images and Caustics . ...... 39 7.6.1 T2 Refractive-Index Model of Gravitational Lensing . 39 7.6.2 T2 LensingbyaPointMass . .. .. 40 7.6.3 T2 LensingofaQuasarbyaGalaxy . 42 7.7 Polarization .................................... 46 7.7.1 Polarization Vector and its Geometric-Optics PropagationLaw. 47 7.7.2 T2 GeometricPhase .......................... 48 i Part III OPTICS ii Optics Version 1207.1.K.pdf, 28 October 2012 Prior to the twentieth century’s quantum mechanics and opening of the electromagnetic spectrum observationally, the study of optics was concerned solely with visible light. Reflection and refraction of light were first described by the Greeks and further studied by medieval scholastics such as Roger Bacon (thirteenth century), who explained the rain- bow and used refraction in the design of crude magnifying lenses and spectacles.
    [Show full text]
  • Coulomb's Law Thu January 12, 2017 Ch16
    Matching Objectives from AAMC report Scientific Foundations for Future Week Day Date Reading Content MCAT Content Categories Physicians 4C. Electrostatics:: Charge, conductors, charge conservation 1 Tue January 10, 2017 Ch16 - Electrostatics I 16.1-16.3 Electric Charge/ Coulomb's Law 4C. Electrostatics:: Insulators E1-2c. Use spatial reasoning to interpret multidimensional numerical and visual data (e.g., protein structure or geographic information or electric/magnetic 4C. Electrostatics:: Electric field: field lines Thu January 12, 2017 Ch16 - Electrostatics I 16.4 Electric Field fields). 4C. Electrostatics:: Electric field due to charge distribution E4-3a. Distinguish between ionic interactions, van derWaals interactions, hydrogen bonding, and hydrophobic interactions. Electric Field due to charge 2 Tue January 17, 2017 Ch16 - Electrostatics I 16.5 4C. Electrostatics:: Electric field due to charge distribution distributions Thu January 19, 2017 Ch16 - Electrostatics I 16.6 Gauss' Law 3 Tue January 24, 2017 Ch16 - Electrostatics I 16.7 Gauss' Law 4C. Electrostatics:: Electric field due to charge distribution Thu January 26, 2017 Ch17 - Electrostatics II 17.1-17.3 Electric Potential, Equipotential 4C. Electrostatics:: Potential difference, absolute potential at a location 4C. Circuit Elements:: Capacitance 4C. Circuit Elements:: Parallel plate capacitor Capacitors, Capacitor Combinations 4C. Circuit Elements:: Dielectrics 4 Tue January 31, 2017 Ch17 - Electrostatics II 17.4-17.7 and Dielectrics 4C. Circuit Elements:: Capacitors in series 4C. Circuit Elements:: Capacitors in parallel 4C. Circuit Elements:: Energy of charged capacitor 4C. Circuit Elements:: Current, sign conventions, units E3-2c. Apply understanding of electrical principles to the hazards of electrical Thu February 2, 2017 Ch18 - Moving Charges 18.1-18.3 Current and Resistance 4C.
    [Show full text]
  • Geometrical Optics Notation & Sign Conventions
    EELE 481/582 Optical Design Montana State University – S15 J. A. Shaw Geometrical Optics Notation & Sign Conventions Physics textbook convention (e.g. Hecht’s Optics) … not used in this class Object distances are positive to the left, negative to the right Image distances are negative to the left, positive to the right Optical engineering convention …what we will use Directed distances are positive to the right, negative to the left Angles are positive when they have positive slope relative to the local optical axis and negative when they have negative slope relative to the local optical axis. Curvature is positive when the center of curvature lies to the right Curvature is negative when the center of curvature lies to the left Index of refraction changes sign on reflection (e.g., air index = -1 after reflection) Effective focal length is positive if initially parallel rays are caused to converge; effective focal length is negative if initially parallel rays are caused to diverge Greivenkamp (Field Guide to Geometrical Optics) Thin lenses object n f n΄ image space space h (+) u (+) u΄ (-) h΄ (-) z (-) z΄ (+) L h' z' u Transverse magnification in air m h z u' n' Longitudinal magnification m m2 n 1 1 1 Image & object locations z' z f 1 Effective focal length f f EFL ( = optical power) E (sign denotes converging/diverging) Page 1 of 5 EELE 481/582 Optical Design Montana State University – S15 J. A. Shaw n Front focal length (directed distance) f nf F E n' Rear focal length (directed distance) f ' n' f R E f f ' Effective focal length from front or rear focal lengths f F R E n n' ☼ Note: the front and rear focal lengths are directed distances (signs determined by direction), but the effective focal length is not (its sign is determined by whether the surface or element causes incoming parallel rays to converge or diverge).
    [Show full text]
  • Chapter 3 Geometrical Optics (A.K.A. Ray Optics
    Chapter 3 Geometrical Optics (a.k.a. Ray Optics 3.1 Wavefront Geometrical optics is based on the wave theory of light, and it may be though of a tool that explain the behavior of light, and helps to predict what will happen with light in different situations. It's also the basis of constructing many types of optical devices, such as, e.g., photographic cameras, micro- scopes, telescopes, fiber light guides, and many others. Geometric optics is an extensive field, there is enough material in it to “fill” an entire academic course, even a two-term course. In Ph332, out of necessity, we can only devote to this material three class hours at the mos, so we have to limit ourselves to the most fundamental aspects of this field of optics. As mentioned, geometrical optics is based on the wave theory of light. We have already talked about waves, but only about the most simple ones, propagating in one direction along a single axis. But this is only a special case { in fact, there is a whole lot of other possible situations. A well- known scenario is a circular wave, which can be excited simply by throwing a stone into quiet lake water, as shown, e.g., in this Youtube clip. There are spherical waves { for instance, if an electronic speaker or any other small device generates a sound, the sound propagates in all direction, forming a spherical acoustic wave. In addition to that, the functions describing more complicated waves are 1 no longer as simple as those we used in Chapter 2, namely: 2π 2π ∆y(t) = A sin x − t λ T In the case of a circular wave spreading over a plane (e.g., on the lake surface), we need two coordinates to describe the position on the plane.
    [Show full text]
  • Foundations of Geometrical Optics Section 1 Introduction
    OPTI-201/202 Geometrical Geometrical Optics OPTI-201/202 and Instrumental 1-1 © Copyright2018JohnE.Greivenkamp Foundations of Geometrical Optics Section 1 Introduction OPTI-201/202 Geometrical Geometrical Optics OPTI-201/202 and Instrumental 1-2 Optics © Copyright2018JohnE.Greivenkamp Optics is the field of science and engineering encompassing the physical phenomena associated with the generation, transmission, manipulation, detection and utilization of light. From National Research Council Report: “Harnessing Light” OPTI-201/202 Geometrical Geometrical Optics OPTI-201/202 and Instrumental 1-3 History of Optics © Copyright2018JohnE.Greivenkamp In the beginning… “Let there be light” ~300 BC Euclid (Alexandria) in his Optica noted that light travels in straight lines and described the law of refraction. He believed that vision involved rays going from the eye to the object. 100-150 BC Hero (Alexandria) studied reflection for a plane mirror and showed that the actual ray path was the shortest of all possible paths. 140 AD Ptolemy (Alexandria) studied refraction and suggested that the angle of refraction was proportional to the angle of incidence. 1000 Alhazen (Bazra) investigated magnification produced by lenses, atmospheric refraction, and spherical and parabolic mirrors. He was aware of spherical aberration. His work was translated into Latin and available to later European scholars. 1267 Roger Bacon (England) considered the speed of light to be finite. He studied the magnification of small objects using convex lenses, and he attributed rainbows to the reflection of sunlight from individual raindrops. 1268-1289 The invention of eyeglasses in Italy. The lenses were made of natural crystal. 1590 Zacharius Jensen (Netherlands) constructed a compound microscope.
    [Show full text]
  • Invisibility Cloaking Based on Geometrical Optics for Visible Light
    J. Europ. Opt. Soc. Rap. Public. 8, 13037 (2013) www.jeos.org Invisibility cloaking based on geometrical optics for visible light H. Ichikawa Faculty of Engineering, Ehime University, 3 Bunkyo, Matsuyama 790-8577, Japan [email protected] M. Oura Artner Co., Ltd., 3-2-18 Nakanoshima, Kita-ku, Osaka 530-0005, Japan T. Taoda Faculty of Engineering, Ehime University, 3 Bunkyo, Matsuyama 790-8577, Japan Optical cloaking has been one of unattainable dreams and just a subject in fiction until recently. Several different approaches to cloaking have been proposed and demonstrated: stealth technology, active camouflage and transformation optics. The last one would be the most formal approach modifying electromagnetic field around an object to be cloaked with metamaterials. While cloaking based on transformation optics, though valid only at single frequency, is experimentally demonstrated in microwave region, its operation in visible spectrum is still distant from realisation mainly owing to difficulty in fabricating metamaterial structure whose elements are much smaller than wavelength of light. Here we show that achromatic optical cloaking in visible spectrum is possible with the mere principle based on geometrical optics. In combining a pair of polarising beam splitters and right-angled prisms, rays of light to be obstructed by an object can make a detour to an observer, while unobstructed rays go straight through two polarising beam splitters. What is observed eventually through the device is simply background image as if nothing exists in between. [DOI: http://dx.doi.org/10.2971/jeos.2013.13037] Keywords: Invisibility cloaking, geometric optical design, polarisation-selective devices, propagating methods 1INTRODUCTION History of invisibility cloaking in real world is not old.
    [Show full text]
  • Basic Geometrical Optics
    FUNDAMENTALS OF PHOTONICS Module 1.3 Basic Geometrical Optics Leno S. Pedrotti CORD Waco, Texas Optics is the cornerstone of photonics systems and applications. In this module, you will learn about one of the two main divisions of basic optics—geometrical (ray) optics. In the module to follow, you will learn about the other—physical (wave) optics. Geometrical optics will help you understand the basics of light reflection and refraction and the use of simple optical elements such as mirrors, prisms, lenses, and fibers. Physical optics will help you understand the phenomena of light wave interference, diffraction, and polarization; the use of thin film coatings on mirrors to enhance or suppress reflection; and the operation of such devices as gratings and quarter-wave plates. Prerequisites Before you work through this module, you should have completed Module 1-1, Nature and Properties of Light. In addition, you should be able to manipulate and use algebraic formulas, deal with units, understand the geometry of circles and triangles, and use the basic trigonometric functions (sin, cos, tan) as they apply to the relationships of sides and angles in right triangles. 73 Downloaded From: https://www.spiedigitallibrary.org/ebooks on 1/8/2019 DownloadedTerms of Use: From:https://www.spiedigitallibrary.org/terms-of-use http://ebooks.spiedigitallibrary.org/ on 09/18/2013 Terms of Use: http://spiedl.org/terms F UNDAMENTALS OF P HOTONICS Objectives When you finish this module you will be able to: • Distinguish between light rays and light waves. • State the law of reflection and show with appropriate drawings how it applies to light rays at plane and spherical surfaces.
    [Show full text]