# Problem with diffraction equations (diffraction and astrophysics)

1. Mar 27, 2014

### IQScience

1. The problem statement, all variables and given/known data
Hey everyone. Just as a preface so you all know, I'm in England studying AQA Physics A.
I'm currently studying the optional astrophysics module.
I have a problem - in all the textbooks and revision guides I have, there is a topic on diffraction of light through a grating (Unit 2 topic), and there's a topic in my Collins Unit 5 textbook on resolving power of telescopes (Unit 5 optional topic astrophysics).

First of all, in the Unit 2 diffraction topic, it states that for light being diffracted, the equation for finding the maxima is given by sinθ = nλ/d (where n = a whole number (integer)).

However, in the Unit 5 astrophysics resolving power of telescopes topic, it states that for light being diffracted (and forming an Airy disc), the equation for finding the minima is given by sinθ = nλ/d (where n = a whole number (integer)).

I'm really confused here, because both are examples of light diffraction and I have no idea whatsoever why the same equation has been used to describe the locations of maxima and minima. It just doesn't make sense.

Help pls?

edit: I should note that in the equation, d = mirror diameter (telescopes) or distance from grating to screen (gratings).

2. Mar 27, 2014

### BvU

Can't find your Airy minima formula: http://en.wikipedia.org/wiki/Airy_disk has quite different numbers...

Anyway, for gratings with sinθ = nλ/d you have a correct formula that applies to a series of equidistant (distance d) line sources. They simply add up constructively if path differences are an integer number times λ.

The Airy formula is for one single extended circle-shaped source (diameter d). Difficult to compare with the above. Basically, all the points of the source (aperture) act as point sources (Huygens principle) and the diffraction pattern is an integral over the source. See Fraunhofer_diffraction, in particular "circular aperture" and "single slit".

In fact, for a comparson with the grating, it's a bit easier to look at the single slit diffraction pattern.

All are Fraunhofer diffraction patterns and they are a beautiful entrance to the world of Fourier transforms. Turns out that a convolution in one domain is a multiplication in the other. So the diffraction pattern of a sequence of slits with a certain width is the product of the sinθ = nλ/d of the centers of the slits at distances d times the diffraction pattern of a single slit with e.g. width d/4.

There's a lot of nice things to explore and play with in hyperphysics on Fraunhofer