Fraunhofer interference and diffraction patterns from single and double slits.
References: Bayman & Hammermesh sec. 4.2; Tipler & Mosca sec. 33.5.

Version 0.2
To do:
(i) Add slits
(ii) Adjust normalization for multiple slits?
(iii) Resolve singularities at multiple slit peaks
(iv) Parameterize wavelength better/differently?

I0 – single-slit intensity; a – slit width; d – slit separation; L – wavelength (as a multiple of d); D – distance to screen (so tanT=x/D gives the position on the screen corresponding to angle T. For small angles note sinT = tanT = x/D.)

I_0=1,D=10

a=slider([0,0.25,40])

d=slider([0,2,40])

L=slider([0,1,40])

Phase difference between rays from top and bottom of each slit as a function of screen position (X):

function(P,X)=2*pi/L*a/D*X

Phase difference between rays from centers of adjacent slits as a function of screen position (X):

function(D,X)=2*pi/L*d/D*X

Intensity for single-slit diffraction pattern:

function(I_1,X)=I_0*[sin(function(P,X)/2)/[function(P,X)/2]]^2

Intensity for two-slit interference-diffraction pattern (note how interference term is modulated by the single-slit diffraction pattern):

function(I_2,X)=4*function(I_1,X)*[cos([function(D,X)/2])]^2

Intensity for two-slit interference pattern (neglecting diffraction):

function(I_(2*I),X)=4*I_0*[cos([function(D,X)/2])]^2

Intensity for m-slit diffraction pattern:

function(I_N,X,m)=function(I_1,X)*[sin([m*function(D,X)/2])/sin([function(D,X)/2])]^2

Envelope for m-slit diffraction:

function(I_E,X,m)=m^2*function(I_1,X)

Location of intensity maxima as a function of position on the screen.

d*x/D=m*L,in(m,set(-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10)),0<y<I_0

To plot as a function of angle replace x->Dsinx:

d*sin(x)=m*L,in(m,set(-5,-4,-3,-2,-1,0,1,2,3,4,5)),0<y<I_0

function(I_1,x)

function(I_2,x)

function(I_(2*I),x)

function(I_N,x,2)

function(I_E,x,2)

Bragg condition. d is now the interplanar spacing.

2*d*sin(x)=m*L,in(m,set(-5,-4,-3,-2,-1,0,1,2,3,4,5)),0<y<I_0



Author: David A. Craig <http://web.lemoyne.edu/~craigda/>

Graph of the formula

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