Diffraction of a circular aperture

In summary, the discussion revolves around the use of mathematical functions, specifically the first order Bessel function (J1) and the Sin(x)/x function, to describe the diffraction patterns produced by circular and rectangular apertures. While the intensity is given by J1(x)/x for a circular aperture, the function is undefined at x=0, causing confusion. However, this is resolved by looking at the limit of the function as x approaches 0. Additionally, the use of Bessel functions is seen as a shift from Fourier-style results to circular coordinates in calculating diffraction patterns. Resources such as advanced optics books can provide more information on this topic.
  • #1
KFC
488
4
It is quite typical example for a text to mention Airy disc (a diffraction patten for a circular aperture), also in wiki http://en.wikipedia.org/wiki/Airy_disc. But what wiki confusing me is , in the mathematical details section, the intensity is given by J1(x)/x, where J1 is the first order Bessel function, it is ZERO around x=0. But the Airy disc has a maximum in the center, so how can on use J1(x)/x to describe an Airy disc?

And I saw some introduction on using a circular slit (annulus) to produce a diffraction pattern and the it is said that annulus will produce a bessel beam on the screen. Any one know if this is true and where can I find some information on this?
 
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  • #2
The intensity should be J0(x)/x: also called a 'sombrero' function.

edit: oops... yep, it's J1(x)/x.
 
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  • #3
KFC said:
It is quite typical example for a text to mention Airy disc (a diffraction patten for a circular aperture), also in wiki http://en.wikipedia.org/wiki/Airy_disc. But what wiki confusing me is , in the mathematical details section, the intensity is given by J1(x)/x, where J1 is the first order Bessel function, it is ZERO around x=0. But the Airy disc has a maximum in the center, so how can on use J1(x)/x to describe an Airy disc?

And I saw some introduction on using a circular slit (annulus) to produce a diffraction pattern and the it is said that annulus will produce a bessel beam on the screen. Any one know if this is true and where can I find some information on this?

Any idea what the diffraction pattern of a rectangular slit is? Hint: It's based on Sin(x)/x. Go look up that function. Notice how when X = 0 that sin (x) = 0? But 0/0 is undefined. Hence one must look at the limit of the sin(x)/x function as x -> 0. It's NOT zero! Go Google the sinx/x function and see what it looks like!

well, when you shift from rectangular coordinates to circular coordinates the calculation of the diffraction patterns shifts from Fourier-style results (with sin and cos) to Bessle function results. There you end up with J1(x)/x which is analogous to the sinx/x above. And the same ideas apply at x = 0. OK?

For more information go look at any advanced optics book such as Born and Wolf.
 

1. What is diffraction of a circular aperture?

Diffraction of a circular aperture is a phenomenon that occurs when light waves pass through a circular opening or aperture. The light waves spread out and interfere with each other, creating a diffraction pattern.

2. How does the size of the aperture affect diffraction?

The smaller the aperture, the more pronounced the diffraction effect will be. This is because smaller apertures cause greater interference between the light waves, resulting in a more complex diffraction pattern.

3. What is the relationship between diffraction and wavelength?

The amount of diffraction that occurs is directly related to the wavelength of the light waves. Longer wavelengths, such as red light, will diffract more than shorter wavelengths, such as blue light.

4. What is the purpose of studying diffraction of a circular aperture?

Studying diffraction of a circular aperture allows scientists to understand the properties of light and how it behaves when passing through small openings. This is important for various applications such as optical instruments and imaging techniques.

5. How is the diffraction pattern affected by the shape of the aperture?

The shape of the aperture can greatly influence the diffraction pattern. A circular aperture will produce a circular diffraction pattern, while a square aperture will produce a square pattern. Other shapes, such as triangles or rectangles, will produce unique diffraction patterns.

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