DOCSLIB.ORG

Today in Physics 218: Diffraction by a Circular Aperture Or Obstacle ‰ Circular-Aperture Diffraction and the Airy Pattern ‰ Circular Obstacles, and Poisson’S Spot

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Today in Physics 218: Diffraction by a Circular Aperture Or Obstacle ‰ Circular-Aperture Diffraction and the Airy Pattern ‰ Circular Obstacles, and Poisson’S Spot Today in Physics 218: diffraction by a circular aperture or obstacle Circular-aperture diffraction and the Airy pattern Circular obstacles, and Poisson’s spot. V773 Tau: AO off V773 Tau: AO on (and brightness Neptune orbit diameter, turned way up) seen from same distance 2 April 2004 Physics 218, Spring 2004 1 The circular aperture Most experimental situations in optics (e.g. telescopes) have circular apertures, so the application of the Kirchhoff integral to diffraction from such apertures is of particular interest. We start with a plane wave incident normally on a circular hole with radius a in an otherwise opaque screen, and ask: what is the distribution of the intensity of light on a screen a distance Ra >> away? The field in the aperture is constant, spatially: −itω ExytEeNN(,,)′′ = 0 , and the geometry is as follows: 2 April 2004 Physics 218, Spring 2004 2 xr′′= cosφ ′ x The circular aperture yr′′= sinφ ′ (continued) da′′′′= rdrdφ da’ Xq= cosΦ φ r’ Yq= sin Φ a X r y R Φ Small angles: θ q X kq cos Φ kk≅= z,Z x rr kq sin Φ k ≅ Y y r 2 April 2004 Physics 218, Spring 2004 3 The circular aperture (continued) Thus, ∞∞ eikr −+ikx′′ ky Ekkt,,= Exyte (,,)′′()xy dxdy ′′ Fxy()λr ∫∫ N −∞ −∞ a eikr = dr′′ r λr ∫ 0 2π −itω ⎛⎞ikr′ q ¥ dEφφφ′′′N0 e exp⎜⎟− () cos cosΦ+ sin sin Φ ∫ ⎝⎠r 0 ikr()−ω t a 2π EeN0 ⎛⎞ikr′ q =−−Φdr′′ r dφφ ′exp⎜⎟ cos() ′ , λrr∫∫ ⎝⎠ 00 2 April 2004 Physics 218, Spring 2004 4 The circular aperture (continued) The aperture is symmetrical about the z axis, so we expect that the answer will be independent of the “screen” azimuthal coordinate Φ; without loss of generality, then, we can take Φ = 0. The integral over φ′ becomes 2π ⎛⎞ikr′ q I =−dφφ′′exp⎜⎟ cos . ∫ ⎝⎠r 0 Don’t try to integrate that directly; it’s a Bessel function of the first kind, order zero: 2π 1 iucos v ⎛⎞kr′ q JuJu00()−= () = e dv ⇒I =2.π J 0⎜⎟ 2π ∫ ⎝⎠r 0 2 April 2004 Physics 218, Spring 2004 5 Flashback: Bessel functions The Bessel function of the first kind, of order m, can be represented by the integral −m 2π i imvu+ cos v Ju()= e() dv. m 2π ∫ 0 Bessel functions of different order are related by the recurrence relation u d ⎡⎤mm m m uJ() u=⇔= uJ−−() u uJ () u vJ () vdv . du ⎣⎦mm11 m∫ m 0 Recurrence relations of special functions are very useful when one has to integrate those special functions, as you’re about to see. 2 April 2004 Physics 218, Spring 2004 6 Flashback: Bessel functions (continued) J (u) 1 0 Zeroes of the Bessel functions J1 (u) J0 J1 J2 J u 0.5 2 ( ) 2.405 0 0 5.520 3.832 5.136 0 8.654 7.016 8.417 11.792 10.174 11.620 0.5 14.931 13.324 14.796 10 50 510 u 2 April 2004 Physics 218, Spring 2004 7 The circular aperture (continued) So the far field is ikr( −ω t) a 2πEeN0 ⎛⎞kqr′ EF () q, t= drrJ′′0 ⎜⎟ λrr∫ ⎝⎠ 0 −ω 2 kaq/ r 2πEeikr() t ⎛⎞r = N0 ⎜⎟ vJ() v dv . λrkq∫ 0 ⎝⎠ 0 Now use the recurrence relation, with m = 1: u uJ10() u= ∫ vJ () v dv ; 0 ikr()−ω t 2 2π EeN0 ⎛⎞r ⎛⎞⎛⎞kaq kaq EqtF (),.= ⎜⎟⎜⎟⎜⎟ J1 λrkqrr⎝⎠⎝⎠⎝⎠ 2 April 2004 Physics 218, Spring 2004 8 The circular aperture (continued) Rearrange the field in a somewhat more convenient form: 2 ikr( −ω t) π aEN0 e r ⎛⎞kaq EqtF (),2,= J1 ⎜⎟ λrkaqr⎝⎠ −ω EAeikr() t 2Jka()θ or Et()θ ,= N0 1 , F λθrka 22 2 c ∗ cEN0 A ⎡⎤2Jka1 ()θ whence IkaFFF()θθθ== E()(), tE , t ⎢⎥ . 8πθ8πλ 22r ⎣⎦ka This leaves a minor problem: the expression is indeterminate at ka θ = 0. But the recurrence relation can help us again: 2 April 2004 Physics 218, Spring 2004 9 The circular aperture (continued) Take the recurrence relation at m = 1 and use the chain rule: d dJ1 uJ011() u=⎡⎣⎦ uJ() u ⎤=+ J () u u() u du du dJ Ju() Ju()=+1 () u 1 . 0 du u Note that JJ01(01) == and (00:) dJ 1 u dJJu() dJ () 1=+11() 0 lim1 =+() 0 lim du duuu→→00 u du 1 dJ = 20,1 () or du Ju() dJ 1 lim1 ==1 () 0 . u→0 udu 2 2 April 2004 Physics 218, Spring 2004 10 The circular aperture (continued) This resolves the indeterminacy: 222 22 cENN00 A⎡⎤1 cE A IF ()02==⎢⎥ , 88πλ22rr⎣⎦2 πλ 22 2 ⎡⎤2Jka1 ()θ IkaIFF()θ = ()0.⎢⎥Airy pattern ⎣⎦kaθ Because J 1 also has zeroes at finite values of ka θ , IkaF ( θ ) has a set of concentric rings for which the intensity is zero (dark rings) The first of these lies at ka θ 1 = 3.832, or 3.832 3.832 λ λ θ == =1.22 . First 1 kaπ 2 a D dark ring 2 April 2004 Physics 218, Spring 2004 11 The Airy pattern Linear scale Logarithmic scale 1 1 0.8 0.1 Ika( θ ) 0.6 I ()0 0.01 0.4 3 10 0.2 4 0 10 5 0 5 10 10 10 5 0 5 10 kaθ 2 April 2004 Physics 218, Spring 2004 12 The Airy pattern (continued) Linear scale Logarithmic scale 2 April 2004 Physics 218, Spring 2004 13 The Airy pattern (continued) A young triple-star system, T Tau, observed at λ = 2.2 µm with adaptive optics on the Palomar 200- inch telescope. The brightest star has saturated the detector in the Airy disk. Note the extensive nest of concentric dark rings around it. (Linear scale.) 2 April 2004 Physics 218, Spring 2004 14 The opaque circular obstacle We can handle the case of diffraction by a circular obstacle quite easily, using the result just obtained. For the field, ikr( −ω t) ∞ 2πEeN0 ⎛⎞kqr′ EF () q, t= drrJ′′0 ⎜⎟ λrr∫ ⎝⎠ a ikr()−ω t ∞ 2πEeN0 ⎛⎞kqr′ = drrJ′′0 ⎜⎟ λrr∫ ⎝⎠ 0 ikr()−ω t a 2πEeN0 ⎛⎞kqr′ − drrJ′′0 ⎜⎟ λrr∫ ⎝⎠ 0 2 ikr()−ω t ikz()−ω t EaeN0π r ⎛⎞kaq =−EeN012, J⎜⎟ λr kaq⎝⎠ r 2 April 2004 Physics 218, Spring 2004 15 The opaque circular obstacle (continued) =−2coskr( 1 z r) and for the intensity, =−2 coskr () 1 cosθ c ∗ Iqt(), = EE 2 2 FFFπ ⎛⎞θ kq 8 ≅=2coskr ⎜⎟ 2cos 2 ⎜⎟ ⎡ ⎛⎞2 ⎝⎠22r ckar2 ⎢ ⎛⎞kaq =+EJN0112⎜⎟⎜⎟ 82π ⎢ ⎜⎟r kaq⎝⎠ r ⎣ ⎝⎠ 2 ka r ⎛⎞kaq −−−⎤ −+2Jeeik() r z ik () r z 1 ⎜⎟( )⎥ rkaq⎝⎠ r ⎦⎥ 2 ⎡⎤⎛⎞222 ckarkar2 ⎢⎥⎛⎞kaq2 ⎛⎞ kaq kq =+EJN0112⎜⎟⎜⎟ − 2cos. J 1 ⎜⎟ 82π ⎢⎥⎜⎟r kaq⎝⎠ r r kaq ⎝⎠ r 2 r ⎣⎦⎝⎠ 2 April 2004 Physics 218, Spring 2004 16 The opaque circular obstacle (continued) This result has the curious property of not being zero inside the shadow of the obstacle. In fact, there’s a sharp peak exactly in the center, with peak intensity ⎡ 222 ⎤ ckaka2 ⎛⎞2 Iqt(),1=+− E ⎢ ⎜⎟ ⎥ . FN82π 0 ⎢ ⎜⎟rr⎥ ⎣ ⎝⎠ ⎦ And there are concentric bright and dark rings that also lie within the shadow, though generally it is much darker there than it is outside the shadow. 2 April 2004 Physics 218, Spring 2004 17 The opaque circular obstacle (continued) Diffraction patterns at λ = 0.635µ m, seen 5 m away from 0.09375 inch, 0.15625 inch, and 0.1875 inch diameter spheres (Ioan Feier, Horst Friedsam and Merrick Penicka, Argonne National Laboratory). 2 April 2004 Physics 218, Spring 2004 18 Poisson’s spot Not all effects named after famous physicists are meant to honor their namesakes. In 1818, the French Academy, led by neo-Newtonians like Laplace, Biot and Poisson, offered a prize for the best work on the theme of diffraction, expecting that the result would be a definitive refutation of the wave theory of light. Fresnel, supported by Ampère and Arago, offered a paper in which he developed the scalar theory of diffraction in much the same way we did, based on the wave theory. During Fresnel’s talk, Poisson pointed out that one of the consequences of Fresnel’s theory was the intensity peak in the center of circular shadows that we just found. 2 April 2004 Physics 218, Spring 2004 19 Poisson’s spot (continued) Poisson did this, of course, because he thought such a result was ridiculous; he meant it as a fatal objection to Fresnel’s theory. But right after the talk, Arago went into his lab, observed the intensity peak and concentric rings in the shadow directly, and proceeded to demonstrate it to the judges. Thus Fresnel was awarded the prize, the corpuscular theory of light stood refuted (until Einstein and Planck came Nick Nicola along), and the intensity peak has been (University of known ever since as Poisson’s spot. Melbourne) 2 April 2004 Physics 218, Spring 2004 20.
Load more

Recommended publications
  • Diffraction Interference Induced Superfocusing in Nonlinear Talbot Effect24,25, to Achieve Subdiffraction by Exploiting the Phases of The
    OPEN Diffraction Interference Induced SUBJECT AREAS: Superfocusing in Nonlinear Talbot Effect NONLINEAR OPTICS Dongmei Liu1, Yong Zhang1, Jianming Wen2, Zhenhua Chen1, Dunzhao Wei1, Xiaopeng Hu1, Gang Zhao1, SUB-WAVELENGTH OPTICS S. N. Zhu1 & Min Xiao1,3 Received 1National Laboratory of Solid State Microstructures, College of Engineering and Applied Sciences, School of Physics, Nanjing 12 May 2014 University, Nanjing 210093, China, 2Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA, 3Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA. Accepted 24 July 2014 We report a simple, novel subdiffraction method, i.e. diffraction interference induced superfocusing in Published second-harmonic (SH) Talbot effect, to achieve focusing size of less than lSH/4 (or lpump/8) without 20 August 2014 involving evanescent waves or subwavelength apertures. By tailoring point spread functions with Fresnel diffraction interference, we observe periodic SH subdiffracted spots over a hundred of micrometers away from the sample. Our demonstration is the first experimental realization of the Toraldo di Francia’s proposal pioneered 62 years ago for superresolution imaging. Correspondence and requests for materials should be addressed to ocusing of a light beam into an extremely small spot with a high energy density plays an important role in key technologies for miniaturized structures, such as lithography, optical data storage, laser material nanopro- Y.Z. (zhangyong@nju. cessing and nanophotonics in confocal microscopy and superresolution imaging. Because of the wave edu.cn); J.W. F 1 nature of light, however, Abbe discovered at the end of 19th century that diffraction prohibits the visualization (jianming.wen@yale. of features smaller than half of the wavelength of light (also known as the Rayleigh diffraction limit) with optical edu) or M.X.
    [Show full text]
  • Introduction to Optics Part I
    Introduction to Optics part I Overview Lecture Space Systems Engineering presented by: Prof. David Miller prepared by: Olivier de Weck Revised and augmented by: Soon-Jo Chung Chart: 1 16.684 Space Systems Product Development MIT Space Systems Laboratory February 13, 2001 Outline Goal: Give necessary optics background to tackle a space mission, which includes an optical payload •Light •Interaction of Light w/ environment •Optical design fundamentals •Optical performance considerations •Telescope types and CCD design •Interferometer types •Sparse aperture array •Beam combining and Control Chart: 2 16.684 Space Systems Product Development MIT Space Systems Laboratory February 13, 2001 Examples - Motivation Spaceborne Astronomy Planetary nebulae NGC 6543 September 18, 1994 Hubble Space Telescope Chart: 3 16.684 Space Systems Product Development MIT Space Systems Laboratory February 13, 2001 Properties of Light Wave Nature Duality Particle Nature HP w2 E 2 E 0 Energy of Detector c2 wt 2 a photon Q=hQ Solution: Photons are EAeikr( ZtI ) “packets of energy” E: Electric field vector c H: Magnetic field vector Poynting Vector: S E u H 4S 2S Spectral Bands (wavelength O): Wavelength: O Q QT Ultraviolet (UV) 300 Å -300 nm Z Visible Light 400 nm - 700 nm 2S Near IR (NIR) 700 nm - 2.5 Pm Wave Number: k O Chart: 4 16.684 Space Systems Product Development MIT Space Systems Laboratory February 13, 2001 Reflection-Mirrors Mirrors (Reflective Devices) and Lenses (Refractive Devices) are both “Apertures” and are similar to each other. Law of reflection: Mirror Geometry given as Ti Ti=To a conic section rot surface: T 1 2 2 o z()U r r k 1 U Reflected wave is also k 1 in the plane of incidence Specular Circle: k=0 Ellipse -1<k<0 Reflection Parabola: k=-1 Hyperbola: k<-1 sun mirror Detectors resolve Images produced by (solar) energy reflected from detector a target scene* in Visual and NIR.
    [Show full text]
  • Cylindrical Waves Guided Waves
    Cylindrical Waves Guided Waves Cylindrical Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648—Waves in Cylindrical Coordinates D. S. Weile Cylindrical Waves 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities Cylindrical Waves Guided Waves Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes D. S. Weile Cylindrical Waves Cylindrical Waves Guided Waves Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes The Scalar Helmholtz Equation Just as in Cartesian coordinates, Maxwell’s equations in cylindrical coordinates will give rise to a scalar Helmholtz Equation. We study it first. r2 + k 2 = 0 In cylindrical coordinates, this becomes 1 @ @ 1 @2 @2 ρ + + + k 2 = 0 ρ @ρ @ρ ρ2 @φ2 @z2 We will solve this by separating variables: = R(ρ)Φ(φ)Z(z) D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes Separation of Variables Substituting and dividing by , we find 1 d dR 1 d2Φ 1 d2Z ρ + + + k 2 = 0 ρR dρ dρ ρ2Φ dφ2 Z dz2 The third term is independent of φ and ρ, so it must be constant: 1 d2Z = −k 2 Z dz2 z This leaves 1 d dR 1 d2Φ ρ + + k 2 − k 2 = 0 ρR dρ dρ ρ2Φ dφ2 z D.
    [Show full text]
  • Sub-Airy Disk Angular Resolution with High Dynamic Range in the Near-Infrared A
    EPJ Web of Conferences 16, 03002 (2011) DOI: 10.1051/epjconf/20111603002 C Owned by the authors, published by EDP Sciences, 2011 Sub-Airy disk angular resolution with high dynamic range in the near-infrared A. Richichi European Southern Observatory,Karl-Schwarzschildstr. 2, 85748 Garching, Germany Abstract. Lunar occultations (LO) are a simple and effective high angular resolution method, with minimum requirements in instrumentation and telescope time. They rely on the analysis of the diffraction fringes created by the lunar limb. The diffraction phenomen occurs in space, and as a result LO are highly insensitive to most of the degrading effects that limit the performance of traditional single telescope and long-baseline interferometric techniques used for direct detection of faint, close companions to bright stars. We present very recent results obtained with the technique of lunar occultations in the near-IR, showing the detection of companions with very high dynamic range as close as few milliarcseconds to the primary star. We discuss the potential improvements that could be made, to increase further the current performance. Of course, LO are fixed-time events applicable only to sources which happen to lie on the Moon’s apparent orbit. However, with the continuously increasing numbers of potential exoplanets and brown dwarfs beign discovered, the frequency of such events is not negligible. I will list some of the most favorable potential LO in the near future, to be observed from major observatories. 1. THE METHOD The geometry of a lunar occultation (LO) event is sketched in Figure 1. The lunar limb acts as a straight diffracting edge, moving across the source with an angular speed that is the product of the lunar motion vector VM and the cosine of the contact angle CA.
    [Show full text]
  • Appendix a FOURIER TRANSFORM
    Appendix A FOURIER TRANSFORM This appendix gives the definition of the one-, two-, and three-dimensional Fourier transforms as well as their properties. A.l One-Dimensional Fourier Transform If we have a one-dimensional (1-D) function fix), its Fourier transform F(m) is de­ fined as F(m) = [ f(x)exp( -2mmx)dx, (A.l.l) where m is a variable in the Fourier space. Usually it is termed the frequency in the Fou­ rier domain. If xis a variable in the spatial domain, miscalled the spatial frequency. If x represents time, then m is the temporal frequency which denotes, for example, the colour of light in optics or the tone of sound in acoustics. In this book, we call Eq. (A.l.l) the inverse Fourier transform because a minus sign appears in the exponent. Eq. (A.l.l) can be symbolically expressed as F(m) =,r 1{Jtx)}. (A.l.2) where ~-~denotes the inverse Fourier transform in Eq. (A.l.l). Therefore, the direct Fourier transform in this case is given by f(x) = r~ F(m)exp(2mmx)dm, (A.1.3) which can be symbolically rewritten as fix)= ~{F(m)}. (A.l.4) Substituting Eq. (A.l.2) into Eq. (A.l.4) results in 200 Appendix A ftx) = 1"1"-'{.ftx)}. (A.l.5) Therefore we have the following unity relation: ,.,... = l. (A.l.6) It means that performing a direct Fourier transform and an inverse Fourier transform of a function.ftx) successively leads to no change. Using the identity exp(ix) = cosx + isinx and Eq.
    [Show full text]
  • The Most Important Equation in Astronomy! 50
    The Most Important Equation in Astronomy! 50 There are many equations that astronomers use L to describe the physical world, but none is more R 1.22 important and fundamental to the research that we = conduct than the one to the left! You cannot design a D telescope, or a satellite sensor, without paying attention to the relationship that it describes. In optics, the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light. The diffraction pattern has a bright region in the center called the Airy Disk. The diameter of the Airy Disk is related to the wavelength of the illuminating light, L, and the size of the circular aperture (mirror, lens), given by D. When L and D are expressed in the same units (e.g. centimeters, meters), R will be in units of angular measure called radians ( 1 radian = 57.3 degrees). You cannot see details with your eye, with a camera, or with a telescope, that are smaller than the Airy Disk size for your particular optical system. The formula also says that larger telescopes (making D bigger) allow you to see much finer details. For example, compare the top image of the Apollo-15 landing area taken by the Japanese Kaguya Satellite (10 meters/pixel at 100 km orbit elevation: aperture = about 15cm ) with the lower image taken by the LRO satellite (0.5 meters/pixel at a 50km orbit elevation: aperture = ). The Apollo-15 Lunar Module (LM) can be seen by its 'horizontal shadow' near the center of the image.
    [Show full text]
  • Simple and Robust Method for Determination of Laser Fluence
    Open Research Europe Open Research Europe 2021, 1:7 Last updated: 30 JUN 2021 METHOD ARTICLE Simple and robust method for determination of laser fluence thresholds for material modifications: an extension of Liu’s approach to imperfect beams [version 1; peer review: 1 approved, 2 approved with reservations] Mario Garcia-Lechuga 1,2, David Grojo 1 1Aix Marseille Université, CNRS, LP3, UMR7341, Marseille, 13288, France 2Departamento de Física Aplicada, Universidad Autónoma de Madrid, Madrid, 28049, Spain v1 First published: 24 Mar 2021, 1:7 Open Peer Review https://doi.org/10.12688/openreseurope.13073.1 Latest published: 25 Jun 2021, 1:7 https://doi.org/10.12688/openreseurope.13073.2 Reviewer Status Invited Reviewers Abstract The so-called D-squared or Liu’s method is an extensively applied 1 2 3 approach to determine the irradiation fluence thresholds for laser- induced damage or modification of materials. However, one of the version 2 assumptions behind the method is the use of an ideal Gaussian profile (revision) report that can lead in practice to significant errors depending on beam 25 Jun 2021 imperfections. In this work, we rigorously calculate the bias corrections required when applying the same method to Airy-disk like version 1 profiles. Those profiles are readily produced from any beam by 24 Mar 2021 report report report insertion of an aperture in the optical path. Thus, the correction method gives a robust solution for exact threshold determination without any added technical complications as for instance advanced 1. Laurent Lamaignere , CEA-CESTA, Le control or metrology of the beam. Illustrated by two case-studies, the Barp, France approach holds potential to solve the strong discrepancies existing between the laser-induced damage thresholds reported in the 2.
    [Show full text]
  • The Laplace Transform of the Modified Bessel Function K( T± M X) Where
    THE LAPLACE TRANSFORM OF THE MODIFIED BESSEL FUNCTION K(t±»>x) WHERE m-1, 2, 3, ..., n by F. M. RAGAB (Received in revised form 13th August 1963) 1. Introduction In the present paper we determine the Laplace transforms of the modified ±m Bessel function of the second kind Kn(t x), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part. The formulae to be established are as follows (l)-(4): f Jo 2m-1 2\-~ -i-,lx 2m 2m 2m J k + v - 2 2m (2m)2mx2 v+1 v + 2 v + 2m (1) 2m 2m 2m where R(k±nm)>0 and x is taken for simplicity to be real and positive" When m = 1, x may be taken to be complex with real part greater than 1. The asterisk in the generalised hypergeometric function denotes that the . 2m . ., v+1 v + 2 v + 2m . factor =— m th e parametert s , , ..., is omitted; m is a positive 2m 2m 2m 2m integer. m m 1 e~"'Kn(t x)dt = 2 ~ V f V J ±l - 2m-l 2 2m 2m 2m 2m J 2m v+1 n v+1. 4p ' 2 2m 2 1m"' (2m)2mx2 2r2m+l v+1 v+2 v + 2m (2) _ 2m 2m ~lm~ E.M.S.—Y Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 05:34:47, subject to the Cambridge Core terms of use. 326 F. M. RAGAB where m is a positive integer, R(± mn +1)>0 and R(p) > 0; x is taken to be rea and positive and the asterisk has the same meaning as before.
    [Show full text]
  • Non-Classical Integrals of Bessel Functions
    J. Austral. Math. Soc. (Series E) 22 (1981), 368-378 NON-CLASSICAL INTEGRALS OF BESSEL FUNCTIONS S. N. STUART (Received 14 August 1979) (Revised 20 June 1980) Abstract Certain definite integrals involving spherical Bessel functions are treated by relating them to Fourier integrals of the point multipoles of potential theory. The main result (apparently new) concerns where /,, l2 and N are non-negative integers, and r, and r2 are real; it is interpreted as a generalized function derived by differential operations from the delta function 2 fi(r, — r^. An ancillary theorem is presented which expresses the gradient V "y/m(V) of a spherical harmonic function g(r)YLM(U) in a form that separates angular and radial variables. A simple means of translating such a function is also derived. 1. Introduction Some of the best-known differential equations of mathematical physics lead to Fourier integrals that involve Bessel functions when, for instance, they are solved under conditions of rotational symmetry about a centre. As long as the integrals are absolutely convergent they are amenable to the classical analysis of Watson's standard treatise, but a calculus of wider serviceability can be expected from the theory of generalized functions. This paper identifies a class of Bessel-function integrals that are Fourier integrals of derivatives of the Dirac delta function in three dimensions. They arise in the practical context of manipulating special functions-notably when changing the origin of the polar coordinates, for example, in order to evaluate so-called two-centre integrals and ©Copyright Australian Mathematical Society 1981 368 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • Radial Functions and the Fourier Transform
    Radial functions and the Fourier transform Notes for Math 583A, Fall 2008 December 6, 2008 1 Area of a sphere The volume in n dimensions is n n−1 n−1 vol = d x = dx1 ··· dxn = r dr d ω. (1) Here r = |x| is the radius, and ω = x/r it a radial unit vector. Also dn−1ω denotes the angular integral. For instance, when n = 2 it is dθ for 0 ≤ θ ≤ 2π, while for n = 3 it is sin(θ) dθ dφ for 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. The radial component of the volume gives the area of the sphere. The radial directional derivative along the unit vector ω = x/r may be denoted 1 ∂ ∂ ∂ ωd = (x1 + ··· + xn ) = . (2) r ∂x1 ∂xn ∂r The corresponding spherical area is ωdcvol = rn−1 dn−1ω. (3) Thus when n = 2 it is (1/r)(x dy−y dx) = r dθ, while for n = 3 it is (1/r)(x dy dz+ y dz dx + x dx dy) = r2 sin(θ) dθ dφ. The divergence theorem for the ball Br of radius r is thus Z Z div v dnx = v · ω rn−1dn−1ω. (4) Br Sr Notice that if one takes v = x, then div x = n, while x · ω = r. This shows that n n times the volume of the ball is r times the surface areaR of the sphere. ∞ z −t dt Recall that the Gamma function is defined by Γ(z) = 0 t e t . It is easy to show that Γ(z + 1) = zΓ(z).
    [Show full text]
  • Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions
    Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions Banu Yılmaz YAS¸AR and Mehmet Ali OZARSLAN¨ Eastern Mediterranean University, Department of Mathematics Gazimagusa, TRNC, Mersin 10, Turkey E-mail: [email protected]; [email protected] Abstract In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel- Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, C¸etinkaya A, Kıymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin transform are obtained. Besides, the unified Bessel, modified Bessel, spherical Bessel and Bessel- Clifford functions are given as a series of Bessel functions. Furthermore, the derivatives, recurrence relations and partial differential equation of the so-called unified family are found. Moreover, the Mellin transform of the products of the unified Bessel functions are obtained. Besides, a three-fold integral representation is given for unified Bessel function. Some of the results which are obtained in this paper are new and some of them coincide with the known results in special cases. Key words Generalized Pochhammer symbol; generalized Bessel function; generalized spherical Bessel function; generalized Bessel-Clifford function 2010 MR Subject Classification 33C10; 33C05; 44A10 1 Introduction Bessel function first arises in the investigation of a physical problem in Daniel Bernoulli's analysis of the small oscillations of a uniform heavy flexible chain [26, 37]. It appears when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates.
    [Show full text]
  • Special Functions
    Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS table of content Chapter 5 Special Functions 5.1 Heaviside step function - filter function 5.2 Dirac delta function - modeling of impulse processes 5.3 Sine integral function - Gibbs phenomena 5.4 Error function 5.5 Gamma function 5.6 Bessel functions 1. Bessel equation of order ν (BE) 2. Singular points. Frobenius method 3. Indicial equation 4. First solution – Bessel function of the 1st kind 5. Second solution – Bessel function of the 2nd kind. General solution of Bessel equation 6. Bessel functions of half orders – spherical Bessel functions 7. Bessel function of the complex variable – Bessel function of the 3rd kind (Hankel functions) 8. Properties of Bessel functions: - oscillations - identities - differentiation - integration - addition theorem 9. Generating functions 10. Modified Bessel equation (MBE) - modified Bessel functions of the 1st and the 2nd kind 11. Equations solvable in terms of Bessel functions - Airy equation, Airy functions 12. Orthogonality of Bessel functions - self-adjoint form of Bessel equation - orthogonal sets in circular domain - orthogonal sets in annular fomain - Fourier-Bessel series 5.7 Legendre Functions 1. Legendre equation 2. Solution of Legendre equation – Legendre polynomials 3. Recurrence and Rodrigues’ formulae 4. Orthogonality of Legendre polynomials 5. Fourier-Legendre series 6. Integral transform 5.8 Exercises Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS Introduction In this chapter we summarize information about several functions which are widely used for mathematical modeling in engineering. Some of them play a supplemental role, while the others, such as the Bessel and Legendre functions, are of primary importance. These functions appear as solutions of boundary value problems in physics and engineering.
    [Show full text]