Problem with diffraction equations (diffraction and astrophysics)

In summary, the person is studying astrophysics and has encountered confusion between the equations for diffraction of light through a grating (sinθ = nλ/d) and the resolving power of telescopes (sinθ = nλ/d). They are seeking clarification on why the same equation is used for both situations and are referred to explore the concept of Fraunhofer diffraction, which involves the use of Fourier transforms. They are encouraged to explore this further on a website called hyperphysics.
  • #1
IQScience
6
0

Homework Statement


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).
 
Physics news on Phys.org
  • #2
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
 

FAQ: Problem with diffraction equations (diffraction and astrophysics)

1. What is diffraction and how does it relate to astrophysics?

Diffraction is the bending and spreading of waves as they pass through an opening or around an obstacle. In astrophysics, this phenomenon is important when studying light from distant celestial objects, as it affects the clarity and resolution of images obtained by telescopes.

2. How do diffraction equations work?

Diffraction equations are mathematical formulas that describe the behavior of waves as they encounter obstacles or openings. These equations take into account factors such as the wavelength of the wave, the size of the opening, and the distance between the source and the observer.

3. What are some common problems with diffraction equations?

One common problem with diffraction equations is that they can become quite complex, especially when dealing with multiple sources or obstacles. Another issue is that they may not accurately predict the behavior of waves in all situations, as they are based on simplified assumptions.

4. How can diffraction equations be applied in astrophysics?

Diffraction equations are used in astrophysics to help astronomers understand and analyze the light from celestial objects. By taking into account the effects of diffraction, scientists can improve the quality of images obtained by telescopes and gain a better understanding of the behavior of light in the universe.

5. Are there any alternative methods for studying diffraction in astrophysics?

Yes, there are alternative methods for studying diffraction in astrophysics, such as using computer simulations or conducting experiments in controlled environments. These methods can provide additional insights and help validate the predictions of diffraction equations.

Back
Top