## Friday, June 3, 2022

### A First Look At Fresnel Diffraction (Part 3)

From examining a circular aperture and the associated Fresnel diffraction we move now to a rectangular aperture - for which analysis we will use the geometry shown below:

Here the standard Cartesian system of coordinates (x,y) has just been applied to the aperture.  From this we can see: R 2   =  Ö x 2 +   2

and also,  r  +  r'  =  h  + h'  +   1/ 2L  ( x 2 +   2 )

Where L is the equivalent focal length or L  =  R 1 2 / l

Where  is the radius of the first Fresnel zone and l  is the associated wavelength used. As with the circular aperture (see Part 2 ) we assume the obliquity factor:

cos (n, r) - cos (n, r')

is present and varies.  Also that the radial factor,  1/ rr' varies slowly compared to the exponential factor, e  i(kr w t  .

The Fresnel-Kirchoff equation we used in Part 2 can then be written:

U p  =  c  ò x2 x1  ò y2 y1    exp (ik (x 2 +   2/2L) dx   dy

=  c  ò x2 x1  exp (kx 2 /2L) dx  ò y2 y1  exp (ky 2 /2L) dy

Where  c  includes all other factors.  For the above integrals it is customary - and easier - to introduce two dimensionless variables, u and v, such that:

u  =  x Ö k/ p L    v =   y Ö k/ p L

We can then recast the integrals:

U p  = c  ò u2 u1  exp (i u 2) /2 du ò v2 v1  exp (i v 2)/2  dv

Where  U 1  =  c  p L  / k

The integrals can be evaluated from the general form:

ò po   exp (i pw 2) /2 dw  =  c (s) +  iS(s)

For which the real and imaginary parts are given by:

c (s) =  ò po   cos (pw 2) /2  dw

S(s) =  ò po   sin (pw 2) /2  dw

then the Cartesian coordinate of any point (x,y) on Cornu's spiral become:

x=  ò  cos ( pv 2 /2)  dv

y =  ò  sin ( pv 2 /2)  dv

The pair are known as Fresnel integrals and a partial table is given below for reference:

The specific graph of c (s)  vs. S(s) is shown below:

and the limits s1 and s2 are marked on the spiral.   A straight line segment drawn from s1 to s2 then gives the value of the integral:

ò s2 s1  exp (ipw 2) /2  dw

(s2 - s1)  = u2 - u1 = (x2 - x1) Ö 2 / l L

For the x -dimension, and:

(s2 - s1)  = v2 - v1 = (y2 - y1) Ö 2 / l L

For the y-dimension.

An alternative form of the spiral  for (x,y) axes is shown below:

When they are added: (1/ Ö2   +   1/ Ö2)  =  / Ö2

The sum is then squared to obtain the intensity of the whole wave, e.g.

I o  =  (/ Ö2) 2   =   4 /2 = 2

I    »   A 2   =   (D x) 2  +   (D y) 2

Suggested Problems:

2) A student is given the top section of a Cornu spiral to analyze for an exam:

Using this, obtain the relative amplitude for the particular diffraction pattern.  From this find the relative intensity.

3) Plot the graph for a Cornu spiral for a single slit diffraction pattern at intervals of D= -0.10 to 3.0 and Dx from 0 to 0.90. On the graph draw a chord from x = 0.4 to x = 0.75 and from this estimate Dalong the Cornu arc.  Hence or otherwise obtain the relative amplitude and the intensity.