The Young Double-Slit Experiment

[PFuser1232]

I have basically two doubts regarding diffraction and interference. First of all, according to my CIE A level Physics book, while the double slit experiment results in fringes that are more or less equally separated, a diffraction grating results in fringes that are not equally separated. Why is this the case? My second doubt is about passing white light through two slits or, any number of slits. What would be observed? Series of white light maximas and in place of dark fringes a series of spectra? (The 7 constituent colours).

 

[UltrafastPED]

See http://h2physics.org/?cat=49

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

 

[dauto]

You mean you have two questions regarding diffraction and interference.

1st question: When talking about the fringes of the double slit experiment we are considering the interference between two paths that go through different slits and are quite different from each other. Only two paths need to be considered. When talking about diffraction fringes we are considering the interference between many slightly different paths that go through the same slit. The math here involves solving an integral - as opposed to simply adding two waves. These are physically quite different situations and there should be no surprise that they lead to quite different behaviors.

2nd question: If white light is used for those experiments, the central fringe will remain unchanged but the side fringes will smear into a rainbow pattern as you described. The further from the center a fringe is the more spreading one observes. This property is put to good use in spectrograph where a diffraction grating is often used to spread a light source into its spectral constituents in order to analyze the light source.

 

[jtbell]

while the double slit experiment results in fringes that are more or less equally separated, a diffraction grating results in fringes that are not equally separated. Why is this the case?

Assuming we are using monochromatic light (single wavelength), in both cases the fringes are equally spaced in terms of sin θ, as can be seen from the equation nλ = d sin θ.

If you view the fringes on a flat screen the distance of a fringe from the center of the screen is x = L tan θ where L is the distance from the slits or grating to the screen. I leave it as an exercise to work out a formula for x in terms of n. You'll find that in general, the spacing between values of x for successive values of n is not equal. However, if x (or θ) is small, the spacing is very nearly equal.

Double slit experiments usually have relatively large values of d (the slit spacing), and you usually have many closely-spaced fringes. Near the center of the pattern they are practically equally spaced. Far from the center they start to "spread out".

With diffraction gratings d is usually much smaller, and the values of x are larger, so you get fewer fringes, spaced further apart, and the difference in spacing becomes easily evident.

 

[PFuser1232]

You mean you have two questions regarding diffraction and interference.

1st question: When talking about the fringes of the double slit experiment we are considering the interference between two paths that go through different slits and are quite different from each other. Only two paths need to be considered. When talking about diffraction fringes we are considering the interference between many slightly different paths that go through the same slit. The math here involves solving an integral - as opposed to simply adding two waves. These are physically quite different situations and there should be no surprise that they lead to quite different behaviors.

2nd question: If white light is used for those experiments, the central fringe will remain unchanged but the side fringes will smear into a rainbow pattern as you described. The further from the center a fringe is the more spreading one observes. This property is put to good use in spectrograph where a diffraction grating is often used to spread a light source into its spectral constituents in order to analyze the light source.

Does that mean that a white maximum would only be observed at the centre? And the rest of the screen would be basically showing repeating rainbow patterns?

 

[jtbell]

Look at the picture near the bottom of the page, in UltrafastPED's first link.

 

[dauto]

Assuming we are using monochromatic light (single wavelength), in both cases the fringes are equally spaced in terms of sin θ, as can be seen from the equation nλ = d sin θ.

If you view the fringes on a flat screen the distance of a fringe from the center of the screen is x = L tan θ where L is the distance from the slits or grating to the screen. I leave it as an exercise to work out a formula for x in terms of n. You'll find that in general, the spacing between values of x for successive values of n is not equal. However, if x (or θ) is small, the spacing is very nearly equal.

Double slit experiments usually have relatively large values of d (the slit spacing), and you usually have many closely-spaced fringes. Near the center of the pattern they are practically equally spaced. Far from the center they start to "spread out".

With diffraction gratings d is usually much smaller, and the values of x are larger, so you get fewer fringes, spaced further apart, and the difference in spacing becomes easily evident.

He said diffraction grating but I think he meant single slit diffraction. The question makes more sense that way. May be the OP should clarify that point?

 

[dauto]

Does that mean that a white maximum would only be observed at the centre? And the rest of the screen would be basically showing repeating rainbow patterns?

Yes, and the further away from the center the more spread out the rainbows become. They might even start running into each other. Each rainbow might spread so much that they would overlap the neighbor rainbow creating a big mess.

 

[Ivin]

I've linked a paper I've written that quantitatively describes how the fringes of the double slit experiment spread out as distance from the center of the screen increases.

http://vixra.org/abs/1412.0163