Plotting a Bessel Function for Diffraction (Fraunhofer)

In summary, the diffraction pattern should result in something like this but when I plot it I get the central peak without the ripples (even when broadening the view).
  • #1
PhDeezNutz
694
441
Homework Statement
I want to plot formula 10.119 in Jackson

##\frac{dP}{d \Omega} \approx P_i \frac{\left(ka\right)^2}{4 \pi}\frac{\left(1 + \cos \theta \right)}{2}\left|\frac{J_1 \left( ka \sin \theta \right)}{ka \sin \theta} \right|^2##

(I have let ##\alpha = 0## for normal incidence)
Relevant Equations
See above
From my understanding of diffraction pattern is supposed to result in something like this

Image 6-14-20 at 10.47 AM (1).jpg


However when I plot it I get the central peak without the ripples (even when broadening the view). My result

Image 6-15-20 at 9.57 AM.jpg


My code is as follows %1) Define the grid. Define vectors so that they include 0, otherwise entire planes are excluded from the picture. We want only to exclude the% origin. Unfortunately there is no mechanism in place to ensure this with arbitrary choices. n = 100; rmax = 5000; x = linspace(-rmax,rmax,n); y = linspace(-rmax,rmax,n); z = linspace(-rmax,rmax,n); %2) Form a meshgrid and the first radial array [X,Y] = meshgrid(x,y); r = sqrt(X.^2 + Y.^2 + (z(90)*ones(size(Y))).^2); rnegative1 = r.^(-1); rnegative2 = r.^(-2); rnegative3 = r.^(-3); rnegative4 = r.^(-4); rnegative5 = r.^(-5);rnegative1(~isfinite(rnegative1)) = 0; rnegative2(~isfinite(rnegative2)) = 0; rnegative3(~isfinite(rnegative3)) = 0; rnegative4(~isfinite(rnegative4)) = 0; rnegative5(~isfinite(rnegative5)) = 0;r2 = sqrt(X.^2 + Y.^2); r2negative1 = r2.^(-1); r2negative2 = r2.^(-2); r2negative3 = r2.^(-3); r2negative4 = r2.^(-4); r2negative5 = r2.^(-5); r2negative1(~isfinite(r2negative1)) = 0; r2negative2(~isfinite(r2negative2)) = 0; r2negative3(~isfinite(r2negative3)) = 0; r2negative4(~isfinite(r2negative4)) = 0; r2negative5(~isfinite(r2negative5)) = 0;%3) Create constants k = 0.001*5*pi; epsilon = 8.85*(10^(-12)); c = 3*((10)^(8)); c1 = (4*pi)^(-1); c2= (4*pi*epsilon)^(-1);mu = (4*pi)*((10)^(-7)); omega = k*sqrt(mu*epsilon); E0 = 1; a = 1;%4) The kasintheta function r2 = sqrt(X.^2 + Y.^2); r3 = sqrt(X.^2 + Y.^2 + (z(90)*ones(size(Y))).^(2)); r3neg1 = r3.^(-1); kasintheta = k*a*r2.*((z(90)).^(-1)); kasinthetaneg1 = kasintheta.^(-1);Jfactor = (besselj(1,kasintheta).*kasinthetaneg1).^2; Jfactor(~isfinite(kasinthetaneg1)) = 0.25;Eratio = 2*((1+ z(90).*r3neg1).^2).*Jfactor; surf(X,Y,Eratio)

The part of the script that reads

Jfactor(~isfinite(kasinthetaneg1)) = 0.25; is to have a central peak instead of a central 0. It is a pathological case because ##J_1## has a zero at zero but ##\frac{J_1 (x)}{x}## has a peak at zero. Analogous to the sinc function.

Again I am getting a central peak but no ripple (not even tiny ones when I broaden my view).

Thanks for any help in advance. I'll probably have a bunch of follow up questions...fair warning :D
 
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  • #2
Image 6-15-20 at 4.14 PM.jpg


I didn't believe the formulas given so I instead I graphed what I thought was reasonable for the poynting vector component going through the back screen.

##S_z = \frac{1}{z} \frac{J_1 \left(ka \sqrt{x^2 + y^2} \right) }{ka \sqrt{x^2 + y^2}}##

I figured ##S_z## had to depend on ##z## and had to decay as ##z## got bigger. Although my solution is qualitatively right I think it's wrong because I also think that the "period" should get bigger as ##z## gets bigger. (i.e. the wave gets more spread out).
 
  • #3
Turns out it is the electric field (z-component flux) that causes the pattern on the back screen.

I'm using some formulas from this paper for a circular aperture.

https://iopscience.iop.org/article/10.1070/PU2002v045n05ABEH001091

Namely the Hertz vector potential at the top of section 6. Then computing the curl twice to get the electric field. Then extracting the z-component.

This is my result.

FFT1possible.jpg


It's not quite there. Getting the middle peak with ripples but the ripples are not cylindrically symmetric. In general the interpolation is "jagged".

Does anyone think if I Fourier Transform this graph I'll get a typical "airy disk pattern"?
 

Related to Plotting a Bessel Function for Diffraction (Fraunhofer)

1. What is a Bessel function?

A Bessel function is a mathematical function that is commonly used in physics and engineering to describe wave phenomena, such as diffraction patterns. It was first introduced by the mathematician Daniel Bernoulli in the 18th century and is named after the mathematician Friedrich Bessel.

2. How is a Bessel function used in diffraction?

In the context of diffraction, a Bessel function is used to describe the intensity of light at different points in the diffraction pattern. It is particularly useful for analyzing diffraction patterns in the Fraunhofer regime, where the distance between the diffracting object and the observation plane is large compared to the size of the diffracting object.

3. What is the mathematical equation for a Bessel function?

The mathematical equation for a Bessel function is given by Jn(x) = ∑m=0 (-1)m (x/2)2m+n / (m! (m+n)!) where n is the order of the Bessel function and x is the variable. There are also other types of Bessel functions, such as Yn(x) and Hn(x), which have different mathematical forms but are related to Jn(x).

4. What is the significance of the Bessel function's order in diffraction?

The order of a Bessel function determines the number and position of the intensity peaks in the diffraction pattern. Higher order Bessel functions have more peaks and their positions are closer together, resulting in a more complex diffraction pattern. The first order Bessel function (n=1) is often used to describe the central maximum in the diffraction pattern, while higher order Bessel functions contribute to the secondary maxima.

5. How can a Bessel function for diffraction be plotted?

To plot a Bessel function for diffraction, one can use mathematical software such as MATLAB or Wolfram Alpha. The Bessel function can be plotted as a function of the variable x, with the order n as a parameter. The resulting plot will show the intensity of the diffraction pattern at different points, with peaks corresponding to the maxima in the pattern. It is also possible to plot multiple Bessel functions with different orders to see their combined effect on the diffraction pattern.

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