Chapter 2 Wave Optics

Total Page:16

File Type:pdf, Size:1020Kb

Chapter 2 Wave Optics EE 485, Winter 2004, Lih Y. Lin Chapter 2 Wave Optics - Improves on Ray Optics by including phenomena such as interference and diffraction. - Limitations: (1) Cannot provide a complete picture of reflection and refraction at the boundaries between dielectric materials. (2) Cannot explain optical phenomena that require vector formalism, such as polarization. 2.1 Postulates of Wave Optics The wave equation c Light speed in a medium: c = 0 (2.1-1) * * n Wave function u(r,t) [position r = (x, y, z) , time t] satisfies 1 ∂ 2u Wave equation ∇2u − = 0 (2.1-2) c2 ∂t 2 Wave* equation* * is linear. Principle of superposition applies: = + u(r,t) u1(r,t) u2 (r,t) Wave equation approximately applicable to media with position-dependent refractive indices, provided that the variation* is slow* within distances of a wavelength → Locally homogeneous, n = n(r), c = c(r). Intensity, power, and energy Optical intensity: optical power per unit area (watts/cm2) * * I(r,t) = 2 u 2 (r,t) (2.1-3) : average over a period >> 1/frequency. Optical power (watts) flowing into an area A normal to the propagation direction: * P(t) = ∫ I(r,t)dA (2.1-4) A Optical energy (joules) collected in a given time interval T is ∫ P(t)dt . T 2.2 Monochromatic Waves * * * u(r,t) = a(r) cos[]2πft + ϕ(r ) (2.2-1) 1 EE 485, Winter 2004, Lih Y. Lin A. Complex Representation and the Helmholtz Equation Complex wavefunction * * * * U (r,t) = a(r) exp[ jϕ(r)]exp( j2πft) = U (r)e jωt (2.2-2,5) * U (r): complex amplitude 1 ∂ 2U ∇2U − = 0 (2.2-4) c2 ∂t 2 The Helmholtz equation and wavenumber Substituting (2.2-5) into (2.2-4) obtains: * (∇2 + k 2 )U (r) = 0 (2.2-7) k = ω : wavenumber (2.2-8) c Optical intensity * * I(r) = U (r) 2 (2.2-10) not a function of time. Wavefronts * The wavefronts are the surfaces of equal phase, ϕ(r) = constant. B. Elementary Waves The plane wave Complex amplitude: * * * U (r) = Aexp(− jk ⋅ r ) = Aexp[− j(k x + k y + k z) (2.2-11) * x y z k*: wavevector → direction of propagation k = k = wavenumber 2 EE 485, Winter 2004, Lih Y. Lin 2π c Wavelength λ = = (2.2-12) k f c is called the phase velocity of the wave. In a medium of refractive index n, f is the same, c λ c = 0 , λ = 0 , k = nk (2.2-14) n n 0 The spherical wave * A − U (r) = e jkr (2.2-15) r * A 2 I(r) = r 2 The paraboloidal wave Fresnel approximation of the spherical wave.* * Examine a spherical wave originating at r = 0 at points r = (x, y, z) sufficiently close to the z axis but far from the origin, so that (x2 + y 2 ) << z . Use paraxial approximation, Taylor series expansion, and Fresnel approximation: * A x2 + y 2 U (r) = exp(− jkz)exp− jk (2.2-16) z 2z First phase term: planar wave. Second phase term: paraboloidal wave. 3 EE 485, Winter 2004, Lih Y. Lin • Validity of Fresnel approximation Fresnel approximation valid for points (x, y) lying within a circle of radius a centered about the z axis at position z, if a satisfies a 4 << 4z3λ , or N θ 2 F m <<1 (2.2-17) 4 θ = where m a / z is the maximum angle, and a 2 N = Fresnel number (2.2-18) F λz 2.4 Simple Optical Components A. Reflection and Refraction Laws of reflection and refraction can be verified by wave optics. Please read the textbook. B. Transmission through Optical Components (Ignore reflection and absorption. Main emphasis on phase shift and associated wavefront bending.) Transmission through a transparent plate • Normal incidence 4 EE 485, Winter 2004, Lih Y. Lin Complex amplitude transmittance = − t(x, y) exp( jnk0d) (2.4-3) d → The plate introduces a phase shift nk d = 2π 0 λ • Oblique incidence = []− θ + θ t(x, y) exp jnk0 (d cos 1 xsin 1 ) Thin transparent plate of varying thickness Transmittance = (Transmittance in air) × (Transmittance in plate) t(x, y) = exp[][]− jk ()d − d(x, y) exp − jnk d(x, y) 0 0 0 (2.4-4) = − []− − exp( jk0d0 )exp j(n 1)k0d(x, y) Thin lens Utilizing Eq. (2.4-4) results in ª x2 + y 2 º t(x, y) = h exp« jk » (2.4-6) 0 ¬ 0 2 f ¼ = − where h0 exp( jnk0d0 ) : constant phase factor R f = : focus length of the lens n −1 5 EE 485, Winter 2004, Lih Y. Lin Diffraction grating An optical component periodically modulates the phase or the amplitude of the incident wave. It can be made of a transparent plate with periodically varying thickness or periodically graded refractive index. The diffraction grating shown above converts an incident plane wave of λ << Λ θ wavelength , traveling at a small angle i with respect to the z axis, into several plane waves at small angles λ θ = θ + q (2.4-9) q i Λ with the z axis. q = 0, ±1, ±2, … : diffraction order In general, without paraxial approximation λ sinθ = sinθ + q (2.4-10) q i Λ 6 EE 485, Winter 2004, Lih Y. Lin C. Graded-index Optical Components Instead of varying thickness, varying refractive index. = []− Varying thickness: t(x, y) exp jn0k0d(x, y) = []− Varying refractive index: t(x, y) exp jn(x, y)k0d0 (2.4-11) 2.5 Interference Linearity of the wave equation → Superposition of the wavefunctions But not superposition of the optical intensity because of interference. (Consider waves of the same frequency in this section.) A. Interference of* Two Waves* * = + U (r) U1(r) U 2 (r) jϕ jϕ ≡ 1 ≡ 2 U1 I1e , U 2 I 2 e Intensity interference equation: I = U 2 = I + I + 2 I I cosϕ 1 2 1 2 (2.5-4,5) ϕ ≡ ϕ −ϕ 2 1 Spatial redistribution of optical intensity. Power conservation still holds. Interferometer Waves superimpose with delay d → ϕ = 2π ()d λ 7 EE 485, Winter 2004, Lih Y. Lin d I = 2I 1+ cos2π (2.5-6) 0 λ Three important examples of intereferometers: Mach-Zehnder interferometer, Michelson interferometer, Sagnac interferometer. 8 EE 485, Winter 2004, Lih Y. Lin d nd nfd Intensity I is a very sensitive function of ϕ = 2π = 2π = 2π λ λ 0 c0 → Can measure small variation of d, n, λ0, or f Interference of two oblique plane waves U = I exp(− jkz) 1 0 = []− θ + θ U 2 I0 exp jk(z cos xsin ) At z = 0, ϕ = kxsinθ , = []+ θ → I 2I0 1 cos(kxsin ) (2.5-7) Period of interference pattern Λ = λ sinθ B. Multiple-wave Interference M waves of equal amplitude and equal phase difference = []− ϕ = U m I0 exp j(m 1) , m 1, 2, ..., M (2.5-9) M = U ∑U m m=1 2 2 sin ()Mϕ 2 I = U = I (2.5-10) 0 sin 2 ()ϕ 2 In the graph of I as a function of ϕ , the number of minor peaks between the main peaks = (M - 1). 9 EE 485, Winter 2004, Lih Y. Lin Infinite number of waves of progressively smaller amplitudes and equal phase difference = (m−1) j(m−1)ϕ = U m I0 r e , m 1, 2, 3, ... r < 1 ∞ I = ∑ = 0 U U m jϕ (2.5-13) m=1 1− re 2 I I = U = max (2.5-15) 1+ ()()2F π 2 sin 2 ϕ / 2 I where I ≡ 0 (2.5-16) max (1− r)2 π r F ≡ : Finesse (2.5-17) 1− r 10 EE 485, Winter 2004, Lih Y. Lin When r approaches 1, I max can be very large! → Principle of optical resonators, lasers. • Physical meaning of F Consider values of ϕ near the ϕ = 0 resonance peak I I ≈ max (2.5-18) 1+ ()F π 2ϕ 2 Full width at half maximum (FWHM) 2π ∆ϕ = (2.5-19) F → Finesse is a measure of the sharpness of the interference function. 2.6 Polychromatic Light Can be expressed as the sum of monochromatic waves over frequency. The pulsed plane wave ∞ * 1 − jkz jωt U (r,t) = ∫ Aω e e dω 2π 0 (2.6-8,9) z = a(t − ) c ∞ 1 jωt a(t) = ∫ Aω e dω (2.6-10) π 2 0 is a function with arbitrary shape σ If a(t) is of finite duration t in time → σ Pulse width in space = c t 11 EE 485, Winter 2004, Lih Y. Lin Pulse width in frequency domain (spectral bandwidth) depends on pulse shape, see Appendix A. E.g., for Gaussian function in space, its Fourier transform is still Gaussian in 1 frequency domain. σ = 1 ps, cσ = 0.3 mm, σ = = 80 GHz. t t f πσ 4 t Interference (beating) between two monochromatic waves j2πf t j2πf t = 1 + 2 U (t) I1e I 2 e (2.6-11) 12 EE 485, Winter 2004, Lih Y. Lin Utilizing the interference equation (2.5-4), = 2 = + + []π ()− I U I1 I 2 2 I1I 2 cos 2 f1 f2 t (2.6-12) − ≡ f1 f2 Beat frequency Interference of M monochromatic waves Consider an odd number M = 2L + 1 waves, each with intensity I0 and frequencies = + fq f0 qf F , q = -L, …, 0, … L << centered about f0 and spaced by f F f0 . L = []π ()+ U (t) I0 ∑exp j2 f0 qf F t (2.6-13) q=−L Utilizing Eq. (2.5-10), 2 2 sin ()Mπt T I = U = I F (2.6-14) 0 2 ()π sin t TF ≡ 2 → A periodic sequence of pulses with period TF 1/ f F , peak intensity M I0 .
Recommended publications
  • The Green-Function Transform and Wave Propagation
    1 The Green-function transform and wave propagation Colin J. R. Sheppard,1 S. S. Kou2 and J. Lin2 1Department of Nanophysics, Istituto Italiano di Tecnologia, Genova 16163, Italy 2School of Physics, University of Melbourne, Victoria 3010, Australia *Corresponding author: [email protected] PACS: 0230Nw, 4120Jb, 4225Bs, 4225Fx, 4230Kq, 4320Bi Abstract: Fourier methods well known in signal processing are applied to three- dimensional wave propagation problems. The Fourier transform of the Green function, when written explicitly in terms of a real-valued spatial frequency, consists of homogeneous and inhomogeneous components. Both parts are necessary to result in a pure out-going wave that satisfies causality. The homogeneous component consists only of propagating waves, but the inhomogeneous component contains both evanescent and propagating terms. Thus we make a distinction between inhomogenous waves and evanescent waves. The evanescent component is completely contained in the region of the inhomogeneous component outside the k-space sphere. Further, propagating waves in the Weyl expansion contain both homogeneous and inhomogeneous components. The connection between the Whittaker and Weyl expansions is discussed. A list of relevant spherically symmetric Fourier transforms is given. 2 1. Introduction In an impressive recent paper, Schmalz et al. presented a rigorous derivation of the general Green function of the Helmholtz equation based on three-dimensional (3D) Fourier transformation, and then found a unique solution for the case of a source (Schmalz, Schmalz et al. 2010). Their approach is based on the use of generalized functions and the causal nature of the out-going Green function. Actually, the basic principle of their method was described many years ago by Dirac (Dirac 1981), but has not been widely adopted.
    [Show full text]
  • Approximate Separability of Green's Function for High Frequency
    APPROXIMATE SEPARABILITY OF GREEN'S FUNCTION FOR HIGH FREQUENCY HELMHOLTZ EQUATIONS BJORN¨ ENGQUIST AND HONGKAI ZHAO Abstract. Approximate separable representations of Green's functions for differential operators is a basic and an important aspect in the analysis of differential equations and in the development of efficient numerical algorithms for solving them. Being able to approx- imate a Green's function as a sum with few separable terms is equivalent to the existence of low rank approximation of corresponding discretized system. This property can be explored for matrix compression and efficient numerical algorithms. Green's functions for coercive elliptic differential operators in divergence form have been shown to be highly separable and low rank approximation for their discretized systems has been utilized to develop efficient numerical algorithms. The case of Helmholtz equation in the high fre- quency limit is more challenging both mathematically and numerically. In this work, we develop a new approach to study approximate separability for the Green's function of Helmholtz equation in the high frequency limit based on an explicit characterization of the relation between two Green's functions and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors. We derive both lower bounds and upper bounds and show their sharpness for cases that are commonly used in practice. 1. Introduction Given a linear differential operator, denoted by L, the Green's function, denoted by G(x; y), is defined as the fundamental solution in a domain Ω ⊆ Rn to the partial differ- ential equation 8 n < LxG(x; y) = δ(x − y); x; y 2 Ω ⊆ R (1) : with boundary condition or condition at infinity; where δ(x − y) is the Dirac delta function denoting an impulse source point at y.
    [Show full text]
  • Waves and Imaging Class Notes - 18.325
    Waves and Imaging Class notes - 18.325 Laurent Demanet Draft December 20, 2016 2 Preface In the margins of this text we use • the symbol (!) to draw attention when a physical assumption or sim- plification is made; and • the symbol ($) to draw attention when a mathematical fact is stated without proof. Thanks are extended to the following people for discussions, suggestions, and contributions to early drafts: William Symes, Thibaut Lienart, Nicholas Maxwell, Pierre-David Letourneau, Russell Hewett, and Vincent Jugnon. These notes are accompanied by computer exercises in Python, that show how to code the adjoint-state method in 1D, in a step-by-step fash- ion, from scratch. They are provided by Russell Hewett, as part of our software platform, the Python Seismic Imaging Toolbox (PySIT), available at http://pysit.org. 3 4 Contents 1 Wave equations 9 1.1 Physical models . .9 1.1.1 Acoustic waves . .9 1.1.2 Elastic waves . 13 1.1.3 Electromagnetic waves . 17 1.2 Special solutions . 19 1.2.1 Plane waves, dispersion relations . 19 1.2.2 Traveling waves, characteristic equations . 24 1.2.3 Spherical waves, Green's functions . 29 1.2.4 The Helmholtz equation . 34 1.2.5 Reflected waves . 35 1.3 Exercises . 39 2 Geometrical optics 45 2.1 Traveltimes and Green's functions . 45 2.2 Rays . 49 2.3 Amplitudes . 52 2.4 Caustics . 54 2.5 Exercises . 55 3 Scattering series 59 3.1 Perturbations and Born series . 60 3.2 Convergence of the Born series (math) . 63 3.3 Convergence of the Born series (physics) .
    [Show full text]
  • Summary of Wave Optics Light Propagates in Form of Waves Wave
    Summary of Wave Optics light propagates in form of waves wave equation in its simplest form is linear, which gives rise to superposition and separation of time and space dependence (interference, diffraction) waves are characterized by wavelength and frequency propagation through media is characterized by refractive index n, which describes the change in phase velocity media with refractive index n alter velocity, wavelength and wavenumber but not frequency lenses alter the curvature of wavefronts Optoelectronic, 2007 – p.1/25 Syllabus 1. Introduction to modern photonics (Feb. 26), 2. Ray optics (lens, mirrors, prisms, et al.) (Mar. 7, 12, 14, 19), 3. Wave optics (plane waves and interference) (Mar. 26, 28), 4. Beam optics (Gaussian beam and resonators) (Apr. 9, 11, 16), 5. Electromagnetic optics (reflection and refraction) (Apr. 18, 23, 25), 6. Fourier optics (diffraction and holography) (Apr. 30, May 2), Midterm (May 7-th), 7. Crystal optics (birefringence and LCDs) (May 9, 14), 8. Waveguide optics (waveguides and optical fibers) (May 16, 21), 9. Photon optics (light quanta and atoms) (May 23, 28), 10. Laser optics (spontaneous and stimulated emissions) (May 30, June 4), 11. Semiconductor optics (LEDs and LDs) (June 6), 12. Nonlinear optics (June 18), 13. Quantum optics (June 20), Final exam (June 27), 14. Semester oral report (July 4), Optoelectronic, 2007 – p.2/25 Paraxial wave approximation paraxial wave = wavefronts normals are paraxial rays U(r) = A(r)exp(−ikz), A(r) slowly varying with at a distance of λ, paraxial Helmholtz equation
    [Show full text]
  • Section 22-3: Energy, Momentum and Radiation Pressure
    Answer to Essential Question 22.2: (a) To find the wavelength, we can combine the equation with the fact that the speed of light in air is 3.00 " 108 m/s. Thus, a frequency of 1 " 1018 Hz corresponds to a wavelength of 3 " 10-10 m, while a frequency of 90.9 MHz corresponds to a wavelength of 3.30 m. (b) Using Equation 22.2, with c = 3.00 " 108 m/s, gives an amplitude of . 22-3 Energy, Momentum and Radiation Pressure All waves carry energy, and electromagnetic waves are no exception. We often characterize the energy carried by a wave in terms of its intensity, which is the power per unit area. At a particular point in space that the wave is moving past, the intensity varies as the electric and magnetic fields at the point oscillate. It is generally most useful to focus on the average intensity, which is given by: . (Eq. 22.3: The average intensity in an EM wave) Note that Equations 22.2 and 22.3 can be combined, so the average intensity can be calculated using only the amplitude of the electric field or only the amplitude of the magnetic field. Momentum and radiation pressure As we will discuss later in the book, there is no mass associated with light, or with any EM wave. Despite this, an electromagnetic wave carries momentum. The momentum of an EM wave is the energy carried by the wave divided by the speed of light. If an EM wave is absorbed by an object, or it reflects from an object, the wave will transfer momentum to the object.
    [Show full text]
  • Properties of Electromagnetic Waves Any Electromagnetic Wave Must Satisfy Four Basic Conditions: 1
    Chapter 34 Electromagnetic Waves The Goal of the Entire Course Maxwell’s Equations: Maxwell’s Equations James Clerk Maxwell •1831 – 1879 •Scottish theoretical physicist •Developed the electromagnetic theory of light •His successful interpretation of the electromagnetic field resulted in the field equations that bear his name. •Also developed and explained – Kinetic theory of gases – Nature of Saturn’s rings – Color vision Start at 12:50 https://www.learner.org/vod/vod_window.html?pid=604 Correcting Ampere’s Law Two surfaces S1 and S2 near the plate of a capacitor are bounded by the same path P. Ampere’s Law states that But it is zero on S2 since there is no conduction current through it. This is a contradiction. Maxwell fixed it by introducing the displacement current: Fig. 34-1, p. 984 Maxwell hypothesized that a changing electric field creates an induced magnetic field. Induced Fields . An increasing solenoid current causes an increasing magnetic field, which induces a circular electric field. An increasing capacitor charge causes an increasing electric field, which induces a circular magnetic field. Slide 34-50 Displacement Current d d(EA)d(q / ε) 1 dq E 0 dt dt dt ε0 dt dq d ε E dt0 dt The displacement current is equal to the conduction current!!! Bsd μ I μ ε I o o o d Maxwell’s Equations The First Unified Field Theory In his unified theory of electromagnetism, Maxwell showed that electromagnetic waves are a natural consequence of the fundamental laws expressed in these four equations: q EABAdd 0 εo dd Edd s BE B s μ I μ ε dto o o dt QuickCheck 34.4 The electric field is increasing.
    [Show full text]
  • Fourier Optics
    Fourier optics 15-463, 15-663, 15-862 Computational Photography http://graphics.cs.cmu.edu/courses/15-463 Fall 2017, Lecture 28 Course announcements • Any questions about homework 6? • Extra office hours today, 3-5pm. • Make sure to take the three surveys: 1) faculty course evaluation 2) TA evaluation survey 3) end-of-semester class survey • Monday are project presentations - Do you prefer 3 minutes or 6 minutes per person? - Will post more details on Piazza. - Also please return cameras on Monday! Overview of today’s lecture • The scalar wave equation. • Basic waves and coherence. • The plane wave spectrum. • Fraunhofer diffraction and transmission. • Fresnel lenses. • Fraunhofer diffraction and reflection. Slide credits Some of these slides were directly adapted from: • Anat Levin (Technion). Scalar wave equation Simplifying the EM equations Scalar wave equation: • Homogeneous and source-free medium • No polarization 1 휕2 훻2 − 푢 푟, 푡 = 0 푐2 휕푡2 speed of light in medium Simplifying the EM equations Helmholtz equation: • Either assume perfectly monochromatic light at wavelength λ • Or assume different wavelengths independent of each other 훻2 + 푘2 ψ 푟 = 0 2휋푐 −푗 푡 푢 푟, 푡 = 푅푒 ψ 푟 푒 휆 what is this? ψ 푟 = 퐴 푟 푒푗휑 푟 Simplifying the EM equations Helmholtz equation: • Either assume perfectly monochromatic light at wavelength λ • Or assume different wavelengths independent of each other 훻2 + 푘2 ψ 푟 = 0 Wave is a sinusoid at frequency 2휋/휆: 2휋푐 −푗 푡 푢 푟, 푡 = 푅푒 ψ 푟 푒 휆 ψ 푟 = 퐴 푟 푒푗휑 푟 what is this? Simplifying the EM equations Helmholtz equation:
    [Show full text]
  • Convolution Quadrature for the Wave Equation with Impedance Boundary Conditions
    Convolution Quadrature for the Wave Equation with Impedance Boundary Conditions a b, S.A. Sauter , M. Schanz ∗ aInstitut f¨urMathematik, University Zurich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland, e-mail: [email protected] bInstitute of Applied Mechanics, Graz University of Technology, 8010 Graz, Austria, e-mail: [email protected] Abstract We consider the numerical solution of the wave equation with impedance bound- ary conditions and start from a boundary integral formulation for its discretiza- tion. We develop the generalized convolution quadrature (gCQ) to solve the arising acoustic retarded potential integral equation for this impedance prob- lem. For the special case of scattering from a spherical object, we derive rep- resentations of analytic solutions which allow to investigate the effect of the impedance coefficient on the acoustic pressure analytically. We have performed systematic numerical experiments to study the convergence rates as well as the sensitivity of the acoustic pressure from the impedance coefficients. Finally, we apply this method to simulate the acoustic pressure in a building with a fairly complicated geometry and to study the influence of the impedance coefficient also in this situation. Keywords: generalized Convolution quadrature, impedance boundary condition, boundary integral equation, time domain, retarded potentials 1. Introduction The efficient and reliable simulation of scattered waves in unbounded exterior domains is a numerical challenge and the development of fast numerical methods ∗Corresponding author Preprint submitted to Computer Methods in Applied Mechanics and EngineeringAugust 18, 2016 is far from being matured. We are interested in boundary integral formulations 5 of the problem to avoid the use of an artificial boundary with approximate transmission conditions [1, 2, 3, 4, 5] and to allow for recasting the problem (under certain assumptions which will be detailed later) as an integral equation on the surface of the scatterer.
    [Show full text]
  • The Human Ear  Hearing, Sound Intensity and Loudness Levels
    UIUC Physics 406 Acoustical Physics of Music The Human Ear Hearing, Sound Intensity and Loudness Levels We’ve been discussing the generation of sounds, so now we’ll discuss the perception of sounds. Human Senses: The astounding ~ 4 billion year evolution of living organisms on this planet, from the earliest single-cell life form(s) to the present day, with our current abilities to hear / see / smell / taste / feel / etc. – all are the result of the evolutionary forces of nature associated with “survival of the fittest” – i.e. it is evolutionarily{very} beneficial for us to be able to hear/perceive the natural sounds that do exist in the environment – it helps us to locate/find food/keep from becoming food, etc., just as vision/sight enables us to perceive objects in our 3-D environment, the ability to move /locomote through the environment enhances our ability to find food/keep from becoming food; Our sense of balance, via a stereo-pair (!) of semi-circular canals (= inertial guidance system!) helps us respond to 3-D inertial forces (e.g. gravity) and maintain our balance/avoid injury, etc. Our sense of taste & smell warn us of things that are bad to eat and/or breathe… Human Perception of Sound: * The human ear responds to disturbances/temporal variations in pressure. Amazingly sensitive! It has more than 6 orders of magnitude in dynamic range of pressure sensitivity (12 orders of magnitude in sound intensity, I p2) and 3 orders of magnitude in frequency (20 Hz – 20 KHz)! * Existence of 2 ears (stereo!) greatly enhances 3-D localization of sounds, and also the determination of pitch (i.e.
    [Show full text]
  • Reflection and Refraction of Light
    Reflection and refraction When light hits a surface, one or more of three things may happen. The light may be reflected back from the surface, transmitted through the material (in which case it will deviate from its initial direction, a process known as refraction) or absorbed by the material if it is not transparent at the wavelength of the incident light. 1. Reflection and refraction Reflection and refraction are governed by two very simple laws. We consider a light wave travelling through medium 1 and striking medium 2. We define the angle of incidence θ as the angle between the incident ray and the normal to the surface (a vector pointing out perpendicular to the surface). For reflection, the reflected ray lies in the plane of incidence, and θ1’ = θ1 For refraction, the refracted ray lies in the plane of incidence, and n1sinθ1 = n2sinθ2 (this expression is called Snell’s law). In the above equations, ni is a dimensionless constant known as the θ1 θ1’ index of refraction or refractive index of medium i. In addition to reflection determining the angle of refraction, n also determines the speed of the light wave in the medium, v = c/n. Just as θ1 is the angle between refraction θ2 the incident ray and the surface normal outside the medium, θ2 is the angle between the transmitted ray and the surface normal inside the medium (i.e. pointing out from the other side of the surface to the original surface normal) 2. Total internal reflection A consequence of Snell’s law is a phenomenon known as total internal reflection.
    [Show full text]
  • Chapter 12: Physics of Ultrasound
    Chapter 12: Physics of Ultrasound Slide set of 54 slides based on the Chapter authored by J.C. Lacefield of the IAEA publication (ISBN 978-92-0-131010-1): Diagnostic Radiology Physics: A Handbook for Teachers and Students Objective: To familiarize students with Physics or Ultrasound, commonly used in diagnostic imaging modality. Slide set prepared by E.Okuno (S. Paulo, Brazil, Institute of Physics of S. Paulo University) IAEA International Atomic Energy Agency Chapter 12. TABLE OF CONTENTS 12.1. Introduction 12.2. Ultrasonic Plane Waves 12.3. Ultrasonic Properties of Biological Tissue 12.4. Ultrasonic Transduction 12.5. Doppler Physics 12.6. Biological Effects of Ultrasound IAEA Diagnostic Radiology Physics: a Handbook for Teachers and Students – chapter 12,2 12.1. INTRODUCTION • Ultrasound is the most commonly used diagnostic imaging modality, accounting for approximately 25% of all imaging examinations performed worldwide nowadays • Ultrasound is an acoustic wave with frequencies greater than the maximum frequency audible to humans, which is 20 kHz IAEA Diagnostic Radiology Physics: a Handbook for Teachers and Students – chapter 12,3 12.1. INTRODUCTION • Diagnostic imaging is generally performed using ultrasound in the frequency range from 2 to 15 MHz • The choice of frequency is dictated by a trade-off between spatial resolution and penetration depth, since higher frequency waves can be focused more tightly but are attenuated more rapidly by tissue The information in an ultrasonic image is influenced by the physical processes underlying propagation, reflection and attenuation of ultrasound waves in tissue IAEA Diagnostic Radiology Physics: a Handbook for Teachers and Students – chapter 12,4 12.1.
    [Show full text]
  • Light Penetration and Light Intensity in Sandy Marine Sediments Measured with Irradiance and Scalar Irradiance Fiber-Optic Microprobes
    MARINE ECOLOGY PROGRESS SERIES Vol. 105: 139-148,1994 Published February I7 Mar. Ecol. Prog. Ser. 1 Light penetration and light intensity in sandy marine sediments measured with irradiance and scalar irradiance fiber-optic microprobes Michael Kiihl', Carsten ~assen~,Bo Barker JOrgensenl 'Max Planck Institute for Marine Microbiology, Fahrenheitstr. 1. D-28359 Bremen. Germany 'Department of Microbial Ecology, Institute of Biological Sciences, University of Aarhus, Ny Munkegade Building 540, DK-8000 Aarhus C, Denmark ABSTRACT: Fiber-optic microprobes for determining irradiance and scalar irradiance were used for light measurements in sandy sediments of different particle size. Intense scattering caused a maximum integral light intensity [photon scalar ~rradiance,E"(400 to 700 nm) and Eo(700 to 880 nm)]at the sedi- ment surface ranging from 180% of incident collimated light in the coarsest sediment (250 to 500 pm grain size) up to 280% in the finest sediment (<63 pm grain me).The thickness of the upper sediment layer in which scalar irradiance was higher than the incident quantum flux on the sediment surface increased with grain size from <0.3mm in the f~nestto > 1 mm in the coarsest sediments. Below 1 mm, light was attenuated exponentially with depth in all sediments. Light attenuation coefficients decreased with increasing particle size, and infrared light penetrated deeper than visible light in all sediments. Attenuation spectra of scalar irradiance exhibited the strongest attenuation at 450 to 500 nm, and a continuous decrease in attenuation coefficent towards the longer wavelengths was observed. Measurements of downwelling irradiance underestimated the total quantum flux available. i.e.
    [Show full text]