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Airy Disc in a microscope

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  1. Sep 24, 2014 #1
    Hi there!
    I wonder where the resolution limit for a microscope comes out. I know that the lens can act as a circular aperture of diameter D and so a point source is diffracted in a disk of angular aperture [tex] 1.22\lambda/D[/tex]
    Two sources are resolved if their distance is greater than (without Abbe correction) [tex]1.22\lambda/NA[/tex]
    How can I obtain this result?
    I'm reading Jenkins and White and they start supposing two point sources, O on the axis of the lens and O' slightly above which form images I and I'. Each image consists of a disk and the angular separation of the disks when they are on the limit of resolution is [tex] 1.22\lambda/D[/tex]
    When this condition holds, the wave from O' diffracted to I has zero intensity and the extreme rays O'BI and O'AI differ in path by 1.22lambda. (B is the top point of the lens, and A is the lower point; I is the position of the image of O and lies on the lens axis) Why do they differ in path by 1.22lambda?

    I also attach the image, taken from the book, of the geometry
    14iq0lw.jpg
     
    Last edited: Sep 24, 2014
  2. jcsd
  3. Sep 25, 2014 #2

    DrDu

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    I suppose you have to derive first the size of the Airy disk in the focal plane. The rest is geometry.
     
  4. Sep 25, 2014 #3
    I tried, but I'm not sure. The distance II' is
    [tex]II'=f\sin(\theta)[/tex]
    where f is the distance of I from the center of the lens.
    The images are resolved if
    [tex]II'\geq f\frac{1.22\lambda}{D}[/tex]
    If we call f' the distance between O and the center of the lens, we have
    [tex]\frac{OO'}{f'}=\frac{II'}{f}[/tex]
    therefore at the limit of the resolution
    [tex] \frac{OO'}{f'}=\frac{f\frac{1.22\lambda}{D}}{f}[/tex]
    But from the geometry we know that
    [tex] f'=\frac{\frac{D}{2}}{\tan(i)}\simeq\frac{D}{2sin(i)}[/tex]
    Thus, in the end
    [tex]OO'=\frac{1.22\lambda}{2\sin(i)}[/tex]

    Is this all right? My doubt is that this derivation is much simpler than that given in textbook.....
     
  5. Sep 25, 2014 #4

    Andy Resnick

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    2016 Award

    It may help to realize that angular resolution and linear resolution are related through the focal length of the lens 'f' and NA = D/2f (approximately).
     
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