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 +   y ²

  and also,  r  +  r'  =  h  + h'  +   1/ 2L  ( x ² +   y 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 +   y 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 p u ²) /2 du ò ^v2 _v1  exp (i p v ²)/2  dv

Where  U ₁  =  c  p L  / k

The integrals can be evaluated from the general form:

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

For which the real and imaginary parts are given by:

  c (s) =  ò ^p_o   cos (pw ²) /2  dw

  S(s) =  ò ^p_o   sin (pw ²) /2  dw  

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

x=  ò ^v ₀  cos ( pv ² /2)  dv

y =  ò ^v ₀  sin ( pv ² /2)  dv

Neither can be integrated in closed form but: 

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   

The projections on the and  iS(s)  and c (s) axes give the real and imaginary parts, respectively.  Any and all problems concerning a rectangular aperture then, will either make use of such a grap and specific limits, or the table of Fresnel integrals. Note here the total arc length is equal to the difference between the limits or s2 - s1 and is proportional to the size of the aperture, i.e.  

(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:

Note also coordinates for Z (½,   ½)  and Z'  (- ½,  - ½),  for the prior version we had (¥),  (- ¥). We can draw a vector straight from 0 to Z or from 0 to Z' and in each case the magnitude is  1/ Ö2.

When they are added: (1/ Ö2   +   1/ Ö2)  =  2 / Ö2

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

I _o  =  (2 / Ö2) ²   =   4 /2 = 2

When working specific problems, I've always advised students taking Calculus- based E&M and Optics to mark off units on the Cartesian Cornu for v in tenths and measure chords on an accurate plot if doing the graph from scratch.  Then using the scale of v on the spiral to obtain the constant length  Dv  of the arc.   

The table of Fresnel integrals can also be used to obtain more accurate results than with the plotted spiral.  Take as an example the interval  Dv  = 0.5.   Read the two values of x at the ends of this interval from the table  and then subtract algebraically to obtain  Dx or the horizontal component of the amplitude.  The same procedure is then used to obtain Dy the vertical component of the amplitude. From these one can then obtain the relative intensity, e.g.:  

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

 Suggested Problems:

1) Without plotting a Cornu spiral, find the hypothetical value of Dv   for such a spiral plotted for the diffraction pattern of a single slit of  width 0.80mm, assuming Fresnel zone parameters a = 40.0 cm and b = 50.0 cm with red light  of 6400 Å.  From this result obtain the relative intensity I.

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 Dv = -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 Dv along the Cornu arc.  Hence or otherwise obtain the relative amplitude and the intensity.