when light passes through a small aperture, a diffraction patern develops as a series of concentic rings. Why isnt the same thing seen with a slit? In the classic double slit experiment, when one uses a single slit to describe the beginning of the experimental setup, they always show a smooth band. What is the typical slit width used in these double slit experimens?
airy disc
- Thread starter bwanaaa
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This looks like a homework problem but I'll anwer quickly anyway. Assuming you're in the Fraunhofer regime (that is your screen is relatively far from the slit) the airy disc is a result of the circular symmetry of the pinhole. To be technical, the airy disc is just the central portion of of a bessel function, which are the concentric rings you see. When you use a single slit, you get a sinc^2 kind of pattern. If you want to generalize for any shape slit, you get its Fourier transform.
would that i had the courage to have pursued my dreams as a physicist, i would know the answer. but the days of homework of this kind, are decades past. the words fraunhofer and bessel have no meaning for me. Fourier transform is a dreamy word that that conjures up a combination of sin waves whose sum is purported to approximate any curve you want. i am just now having the leisure in my life to answer the questions i had written while listening to a boring austrian professor talk about couplets in course 8.022 while carter was president. i know, it sounds weird-why would anyone in his right mind bother to think about these things, but unfortunately, i do.
the basic question that is nagging me is that diffraction causes concentric rings as light passes through a pinhole. but diffraction does not cause bands when light passes through a slit-why? i imagine it has to do with the size of the particle relative to the size of the aperture. I imagine it also has to do with the absolute size as well - airy discs do not occur when basketballs fall through the hoop.
the basic question that is nagging me is that diffraction causes concentric rings as light passes through a pinhole. but diffraction does not cause bands when light passes through a slit-why? i imagine it has to do with the size of the particle relative to the size of the aperture. I imagine it also has to do with the absolute size as well - airy discs do not occur when basketballs fall through the hoop.
Born2bwire
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- Oct 28, 2005
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Going by Raynor's explanation, because Santa knows all, just look at a sinc^2 or bessel function. A sinc function will still have ringing to it, but the amplitude of the ringing is very low, probably not very observable in visible light. If I recall correctly, for a sinc^2, the second maxima is about 14 dB down from the first maxima. A bessel function is a function that describes cylindrical waves. The "ringing" that you get with a bessel function is comparable to the magnitude of the first maxima.
The Fourier transform bit is that if you want to get the general amplitude pattern of the light being emitted by a slit, you take the Fourier transform of the slit. For example, the transform of a rectangule function is a sinc function. I can't actually confirm what Raynor has said off-hand, but if my memory serves me, I think that what he stated is all correct.
As fornax said, there is a definite diffraction pattern when light passes through a slit.
As for the terms, you could probably look it all up on Wikipedia or Mathworld for more formal definitions, but these should get you started.
Fraunhofer regime: your viewing plane is very far away from the slit, relative to the wavelength of light you are using, the Fraunhofer regime is also known as the far-field regime because of this.
Bessel function: Look this up on Mathworld or Wikipedia to see what it looks like but it's basically a circular symmetric pattern of concentric rings. The Airy disc only corresponds to the central part of the pattern.
Fourier transform: Essentially, the Fourier transform shows you the frequency content of a function. In this case, the function is the actually your aperture. It turns out that by looking at the Fraunhofer diffraction pattern, you see the Fourier transform of your aperture! So if you have a slit, the function will look like a sin(x)/x which is also known as the sinc function.
This sinc function (actually sinc^2) is the type of diffraction pattern you would see when using a slit as an apperture. Basically, it would look like a bright band in the center followed by equally sized but progressively dimmer bands as you move away from the center (again, imagine multiplying sin(x) by 1/x).
As an aside, while you can get interference patterns using particles (since quantum-mechanically, particles are waves and vice-versa) and you can also do all the math to figure out diffraction patterns using quantum mechanics, it gets very messy, very quickly. It is much easier to think of simple diffraction problems by using light and considering classical wave theory.
I hope this clears things up a bit for you, but if you would like further details I recommend getting an optics book which will give you a more rigorous and detailed analysis of diffraction.
As for the terms, you could probably look it all up on Wikipedia or Mathworld for more formal definitions, but these should get you started.
Fraunhofer regime: your viewing plane is very far away from the slit, relative to the wavelength of light you are using, the Fraunhofer regime is also known as the far-field regime because of this.
Bessel function: Look this up on Mathworld or Wikipedia to see what it looks like but it's basically a circular symmetric pattern of concentric rings. The Airy disc only corresponds to the central part of the pattern.
Fourier transform: Essentially, the Fourier transform shows you the frequency content of a function. In this case, the function is the actually your aperture. It turns out that by looking at the Fraunhofer diffraction pattern, you see the Fourier transform of your aperture! So if you have a slit, the function will look like a sin(x)/x which is also known as the sinc function.
This sinc function (actually sinc^2) is the type of diffraction pattern you would see when using a slit as an apperture. Basically, it would look like a bright band in the center followed by equally sized but progressively dimmer bands as you move away from the center (again, imagine multiplying sin(x) by 1/x).
As an aside, while you can get interference patterns using particles (since quantum-mechanically, particles are waves and vice-versa) and you can also do all the math to figure out diffraction patterns using quantum mechanics, it gets very messy, very quickly. It is much easier to think of simple diffraction problems by using light and considering classical wave theory.
I hope this clears things up a bit for you, but if you would like further details I recommend getting an optics book which will give you a more rigorous and detailed analysis of diffraction.
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