Scalar theory of diffraction EE 231 Introduction to Optics Scalar theory of diffraction Lesson 5 Andrea Fratalocchi www.primalight.org

The Scalar theory of diffraction Homework 1: How many interference fringes N are found in a disc of radius a? Do the calculation in paraxial and non paraxial approximation. Compare N in the case of R=1m, wavelength 500nm, disc radius a=20mm for paraxial and non paraxial calculations. What is the difference between the two cases?

The Scalar theory of diffraction Homework 1: How many interference fringes N are found in a disc of radius a? Two maxima are separated by an angular distance The number of maxima N in a disc of radius a is therefore:

The Scalar theory of diffraction Homework 1: How many interference fringes N are found in a disc of radius a? Two maxima are separated by an angular distance The number of maxima N in a disc of radius a is therefore: For R=1m, wavelength 500nm and a=20mm, the number of fringes is already N=400!

The Scalar theory of diffraction Homework 1: How many interference fringes N are found in a disc of radius a? Added a phase shift to put in r=0 the point of maximum intensity Exact calculation: The number N of fringes contained in a disc of radius a is then:

The Scalar theory of diffraction Homework 1: How many interference fringes N are found in a disc of radius a? Added a phase shift to put in r=0 the point of maximum intensity Exact calculation: The number N of fringes contained in a disc of radius a is then: With the same numbers as before, we get N=399.96=400. Paraxial approximation gives exactly the same

The Scalar theory of diffraction Propagation of continuum ensembles of waves Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate fields of interest An important case is represented on the left: the propagation of a superposition of plane waves that possess wave vectors lying on a cone of semiaperture At z=0, each plane wave is:

The Scalar theory of diffraction Propagation of continuum ensembles of waves Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate fields of interest At z=0, each plane wave is: In polar coordinates:

The Scalar theory of diffraction Propagation of continuum ensembles of waves Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate fields of interest In polar coordinates: At z=0, each plane wave is:

The Scalar theory of diffraction Propagation of continuum ensembles of waves Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate field of interest Summing up all the contributions: We begin by considering A constant

The Scalar theory of diffraction Propagation of continuum ensembles of waves Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate fields of interest Recalling some results from basic calculus and special functions Bessel function of order n

The Scalar theory of diffraction Propagation of continuum ensembles of waves Bessel function of order n This substitution makes the two integral the same

The Scalar theory of diffraction Propagation of continuum ensembles of waves 0-th order Bessel beam

The Scalar theory of diffraction Propagation of continuum ensembles of waves

The Scalar theory of diffraction Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property?

The Scalar theory of diffraction Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property? 1. Decompose into known waves

The Scalar theory of diffraction Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property? 1. Decompose into known waves

The Scalar theory of diffraction Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property? 1. Decompose into known waves 2. Propagate each wave

The Scalar theory of diffraction Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property? 1. Decompose into known waves 2. Propagate each wave

The Scalar theory of diffraction Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property? 1. Decompose into known waves 2. Propagate each wave 3. Sum them up

The Scalar theory of diffraction Exercise: propagate the following Bessel beam at arbitrary z do you find some interesting property?

The Scalar theory of diffraction Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property? do you find some interesting property? Bessel beams are nondiffracting: intensity is constant along propagation and does not change!

The Scalar theory of diffraction Homework 1: how to experimentally generate a Bessel beam? Homework 2: how to superimpose plane waves and generate high order Bessel beams Jn(kr)? What are their interesting properties? Homework 3: can you think of some possible applications of Bessel beams?

The Scalar theory of diffraction: Bessel Beams References D. McGloin, K. Dholakia, Bessel beams: diffraction in a new light, Contemporary Physics 46 (2005) 15-28. https://www.phys.ksu.edu/reu2014/joshuanelson/BesselBeam1.pdf