Chapter on Diffraction
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The Gauss-Airy Functions and Their Properties
Annals of the University of Craiova, Mathematics and Computer Science Series Volume 43(2), 2016, Pages 119{127 ISSN: 1223-6934 The Gauss-Airy functions and their properties Alireza Ansari Abstract. In this paper, in connection with the generating function of three-variable Hermite polynomials, we introduce the Gauss-Airy function Z 1 3 1 −yt2 zt GAi(x; y; z) = e cos(xt + )dt; y ≥ 0; x; z 2 R: π 0 3 Some properties of this function such as behaviors of zeros, orthogonal relations, corresponding inequalities and their integral transforms are investigated. Key words and phrases. Gauss-Airy function, Airy function, Inequality, Orthogonality, Integral transforms. 1. Introduction We consider the three-variable Hermite polynomials [ n ] [ n−3j ] X3 X2 zj yk H (x; y; z) = n! xn−3j−2k; (1) 3 n j! k!(n − 2k)! j=0 k=0 2 3 as the coefficient set of the generating function ext+yt +zt 1 n 2 3 X t ext+yt +zt = H (x; y; z) : (2) 3 n n! n=0 For the first time, these polynomials [8] and their generalization [9] was introduced by Dattoli et al. and later Torre [10] used them to describe the behaviors of the Hermite- Gaussian wavefunctions in optics. These are wavefunctions appeared as Gaussian apodized Airy polynomials [3,7] in elegant and standard forms. Also, similar families to the generating function (2) have been stated in literature, see [11, 12]. Now in this paper, it is our motivation to introduce the Gauss-Airy function derived from operational relation 2 3 y d +z d n 3Hn(x; y; z) = e dx2 dx3 x : (3) For this purpose, in view of the operational calculus of the Mellin transform [4{6], we apply the following integral representation 1 2 3 Z eys +zs = esξGAi(ξ; y; z) dξ; (4) −∞ Received February 28, 2015. -
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. -
Implementation of Airy Function Using Graphics Processing Unit (GPU)
ITM Web of Conferences 32, 03052 (2020) https://doi.org/10.1051/itmconf/20203203052 ICACC-2020 Implementation of Airy function using Graphics Processing Unit (GPU) Yugesh C. Keluskar1, Megha M. Navada2, Chaitanya S. Jage3 and Navin G. Singhaniya4 Department of Electronics Engineering, Ramrao Adik Institute of Technology, Nerul, Navi Mumbai. Email : [email protected], [email protected], [email protected], [email protected] Abstract—Special mathematical functions are an integral part Airy function is the solution to Airy differential equa- of Fractional Calculus, one of them is the Airy function. But tion,which has the following form [22]: it’s a gruelling task for the processor as well as system that is constructed around the function when it comes to evaluating the d2y special mathematical functions on an ordinary Central Processing = xy (1) Unit (CPU). The Parallel processing capabilities of a Graphics dx2 processing Unit (GPU) hence is used. In this paper GPU is used to get a speedup in time required, with respect to CPU time for It has its applications in classical physics and mainly evaluating the Airy function on its real domain. The objective quantum physics. The Airy function also appears as the so- of this paper is to provide a platform for computing the special lution to many other differential equations related to elasticity, functions which will accelerate the time required for obtaining the heat equation, the Orr-Sommerfield equation, etc in case of result and thus comparing the performance of numerical solution classical physics. While in the case of Quantum physics,the of Airy function using CPU and GPU. -
Fresnel Diffraction.Nb Optics 505 - James C
Fresnel Diffraction.nb Optics 505 - James C. Wyant 1 13 Fresnel Diffraction In this section we will look at the Fresnel diffraction for both circular apertures and rectangular apertures. To help our physical understanding we will begin our discussion by describing Fresnel zones. 13.1 Fresnel Zones In the study of Fresnel diffraction it is convenient to divide the aperture into regions called Fresnel zones. Figure 1 shows a point source, S, illuminating an aperture a distance z1away. The observation point, P, is a distance to the right of the aperture. Let the line SP be normal to the plane containing the aperture. Then we can write S r1 Q ρ z1 r2 z2 P Fig. 1. Spherical wave illuminating aperture. !!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!! 2 2 2 2 SQP = r1 + r2 = z1 +r + z2 +r 1 2 1 1 = z1 + z2 + þþþþ r J þþþþþþþ + þþþþþþþN + 2 z1 z2 The aperture can be divided into regions bounded by concentric circles r = constant defined such that r1 + r2 differ by l 2 in going from one boundary to the next. These regions are called Fresnel zones or half-period zones. If z1and z2 are sufficiently large compared to the size of the aperture the higher order terms of the expan- sion can be neglected to yield the following result. l 1 2 1 1 n þþþþ = þþþþ rn J þþþþþþþ + þþþþþþþN 2 2 z1 z2 Fresnel Diffraction.nb Optics 505 - James C. Wyant 2 Solving for rn, the radius of the nth Fresnel zone, yields !!!!!!!!!!! !!!!!!!! !!!!!!!!!!! rn = n l Lorr1 = l L, r2 = 2 l L, , where (1) 1 L = þþþþþþþþþþþþþþþþþþþþ þþþþþ1 + þþþþþ1 z1 z2 Figure 2 shows a drawing of Fresnel zones where every other zone is made dark. -
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. -
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. -
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. -
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. -
Optics of Gaussian Beams 16
CHAPTER SIXTEEN Optics of Gaussian Beams 16 Optics of Gaussian Beams 16.1 Introduction In this chapter we shall look from a wave standpoint at how narrow beams of light travel through optical systems. We shall see that special solutions to the electromagnetic wave equation exist that take the form of narrow beams – called Gaussian beams. These beams of light have a characteristic radial intensity profile whose width varies along the beam. Because these Gaussian beams behave somewhat like spherical waves, we can match them to the curvature of the mirror of an optical resonator to find exactly what form of beam will result from a particular resonator geometry. 16.2 Beam-Like Solutions of the Wave Equation We expect intuitively that the transverse modes of a laser system will take the form of narrow beams of light which propagate between the mirrors of the laser resonator and maintain a field distribution which remains distributed around and near the axis of the system. We shall therefore need to find solutions of the wave equation which take the form of narrow beams and then see how we can make these solutions compatible with a given laser cavity. Now, the wave equation is, for any field or potential component U0 of Beam-Like Solutions of the Wave Equation 517 an electromagnetic wave ∂2U ∇2U − µ 0 =0 (16.1) 0 r 0 ∂t2 where r is the dielectric constant, which may be a function of position. The non-plane wave solutions that we are looking for are of the form i(ωt−k(r)·r) U0 = U(x, y, z)e (16.2) We allow the wave vector k(r) to be a function of r to include situations where the medium has a non-uniform refractive index. -
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). -
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. -
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.