Diffraction pattern's dependency on wavelength

[blaisem]

Hello. 2 questions:

1. If a diffraction grating is smaller, approaching infinitely smaller, than the wavelength of incident light, what happens to the diffraction pattern? Does the wave still diffract at all?

2.
Wavelength is the distance between two crests of periodic motion. Imagine a wave moving along a horizontal axis that runs into a vertical barrier with a single slit in it. How can the wavelength, a measure of horizontal distance parallel to wave propagation, interact with the length of the slit, a measure of vertical distance? How does the incident crest "know" this ratio of slit length to wavelength in order to diffract accordingly; what tells the incident crest, for example, how far its subsequent crest is away?

If I think of the instantaneous moment a crest meets the slit, and treat that crest as a point source according to the Huygen principle, I don't understand how this point source's subsequent pattern can depend on its distance from the wave behind it. I could freeze the picture at the moment this crest meets the slit, and then adjust its distance to the crest behind it and seemingly arbitrarily alter the outcome of how it diffracts. I don't know of any long-range interaction between water waves, for example, beyond perhaps a force pushing the waves in the direction of propagation -- nothing that would tell the foremost crest how it should diffract when it hits a slit of arbitrary separation.

As an analogy, if I were to view the crests as particles that diffract upon striking the slit, it makes no sense to me that a particle meeting the slit would split up depending on its distance to the particle behind it. What property of this secondary particle is interacting over the distance with the incident particle in order to influence how it will diffract?

In the case of water, it would make more sense to me if the wavelength had some component along the ripple, as that is in the same dimension as the slit itself, rather than an orthogonal component -- for example, if the separation of waves were parallel to the wave propagation.

Can someone provide conceptual insight to how this works? Thank you very much!

 

[UltrafastPED]

The smallest diffraction grating would be the two edges of a scratch. If it is too small (<wavelength) then the wave will not "see" the grating.

The intensity depends upon the number of waves diffracted. This goes down as the number of rulings becomes smaller.

 

[Simon Bridge]

Hello. 2 questions:

1. If a diffraction grating is smaller, approaching infinitely smaller, than the wavelength of incident light, what happens to the diffraction pattern? Does the wave still diffract at all?

In the theory, a diffraction pattern is obtained for an arbitrarily small slit.
However, as the slit gets smaller, the pattern gets fainter.

A narrow grating with the same density and size of slits would eventually be just a single slit.

Understand that you can get diffraction from a single slit, or just an edge. Anything that restricts the possible paths a particular wave vector can follow. It's just that the diffraction grating is set up to make the effect obvious.

A narrow grating with the same overall number of slits has narrower slits spaced closer together.
You can use the equations to work out what that does to the pattern - as the slits get narrower, the diffraction due to individual slits gets more important and the overall pattern gets fainter since it involves less of the wave. Just like the image gets fainter when you make the lens smaller.

There is an issue with matter waves where the slit is bigger than the deBroglie wavelength but smaller than the physical extent of the object. You won't see diffraction of a golf ball for instance, but we can diffract electrons just fine. I think the biggest "particles" that have been diffracted are bucky-balls.

2.
Wavelength is the distance between two crests of periodic motion. Imagine a wave moving along a horizontal axis that runs into a vertical barrier with a single slit in it. How can the wavelength, a measure of horizontal distance parallel to wave propagation, interact with the length of the slit, a measure of vertical distance? How does the incident crest "know" this ratio of slit length to wavelength in order to diffract accordingly; what tells the incident crest, for example, how far its subsequent crest is away?

A water wave knows about the water molecules close by because of the weak attractive forces that make the water a liquid. All waves require some sort of connection between the bits they are made of - otherwise the energy cannot flow.

However - diffraction happens all the time - so the wave does not need much information to know what to do. All it needs to know is that it's path is restricted. The condition that the slit separation has to be comparable to the wavelength is just the condition where the effect becomes big enough and the math simple enough that beginning students have a chance to cope with it.

You don't even need a slit-shape for diffraction - you can look up images for diffraction through a pinhole or a square and see what the effect is.

As an analogy, if I were to view the crests as particles that diffract upon striking the slit, it makes no sense to me that a particle meeting the slit would split up depending on its distance to the particle behind it. What property of this secondary particle is interacting over the distance with the incident particle in order to influence how it will diffract?

The diffraction pattern does not require particles to split up or for them to interact with other particles in the beam.
Quantum mechanical diffraction works differently from that for classical waves in a medium.

Can someone provide conceptual insight to how this works?

Diffraction in water waves can be understood in terms of individual water molecules being weakly attracted to each other and the walls of the slits. As the wave passes through this makes the wavefront bend. This is kinda how a single wide slit gets you circular wavefronts. You get the same effect in a crowd of people having to rush through a small opening.

The diffraction pattern comes from two or more of these circular waves meeting each other.

Note: in water, the wave motion is different from the motion of the water molecules.
The water molecules slosh about an equilibrium position while the wave shape travels.
As they sosh about, they tug on each other, which is how different parts of the wave know how to behave.

Classical waves do not have to be up-down motions.
Sound, for eg. is a variation in pressure (due to back and forth motions).
Classical waves do not have to be in a variable that involves motion (well, not so directly).
Electromagnetic waves are variations in the strength of the electric and magnetic fields and do not even need a medium to travel.

Quantum mechanical diffraction is different.
The best conceptual description I've seen is from Richard Feynman, but it takes a while to wrap your head around.
http://vega.org.uk/video/subseries/8

TLDR: QM tells us nothing about what happens at the slits to get us a diffraction pattern or otherwise.

The diffraction pattern is in the resulting probability of detecting a particle at a particular position. What you detect, though, is individual particles as discrete lumps. Which particle arrives where is random.

The resulting pattern ends up looking similar to the water-wave situation because the equations governing QM are a form of the equations governing water waves. It why we can use the language of waves to describe the behavior of particles in QM.