Explanation for bright fringes in Single Slit Diffraction

[Aurelius120]

TL;DR Summary

The formula for position of bright fringes of Single Slit Fraunhoffer diffraction is given by $$a\sin(\theta_n)=\frac{(2n+1)\lambda}{2}$$
$$\theta_n \approx \sin(\theta_n) \approx \tan(\theta_n)=\frac{x_n}{D}$$
$n=1,2,3,......$
Looking for an intuitive explanation for this formula.
The central bright fringe is brightest. Why?

In Young's Double Slit Experiment, we were shown the complete derivation for location of fringes, width of fringes etc. on interference by two point sources of light and all was well.
In Single Slit Diffraction we were just asked to remember the formulae as they were with little explanation.

I understand that all waves from points equidistant from slit-center on either side interfere constructively at the screen-center but why don't they cancel with waves from points that are in opposite phase? Why are waves from every point interfering constructively with waves from every other point? If there is a combination of both constructive and destructive, why is it brighter than other bright fringes?

A little research gives a clear explanation for dark fringes and why they are formed at path difference of $n\lambda$. For example here.

However I cannot find an explanation for formation of maxima at $\Delta x=\frac{(2n+1)\lambda}{2}$? Is the explanation intuitive or is the reason purely mathematical?(perhaps too complicated to be taught)

 

[BvU]

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/sinslitd.html#c1

 

[Aurelius120]

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/sinslitd.html#c1

Although there is a progressive change in phase as you choose element pairs closer to the centerline, this center position is nevertheless the most favorable location for constructive interference of light from the entire slit and has the highest light intensity if the Fraunhofer diffraction expression is reasonably applicable. If the conditions D >> a and D>> a2/λ are not met for this combination of slit width and screen distance, the Fresnel diffraction result may not have maximum intensity on the centerline.

Still don't understand why it is brightest?

A and B will interfere constructively with each other but destroy C and then other points on either side will destroy whats left. It will be a Central Dark Fringe , Right?

And still don't see why bright fringe at odd multiples of half wavelength

 

[BvU]

I'm looking at the bottom half of lecture notes (Prof. Dmitri Pogosian, Alberta Ca), based on Young and Freedman, 12th ed. (I only have 11th ed, pp 1369...1376) and really don't know what to add.

And there's always the hyperphysics explanation (4 pages, phasors, etc)

In Young's Double Slit Experiment, we were shown the complete derivation for location of fringes, width of fringes etc. on interference by two point sources of light and all was well.

Note that there too the single-slit pattern appears -- as an envelope

$\$

 

[hutchphd]

or Feynman's Lectures. Always a good choice