What Determines the Resolution Limit of a Microscope?

[eoghan]

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 $$1.22\lambda/D$$
Two sources are resolved if their distance is greater than (without Abbe correction) $$1.22\lambda/NA$$
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 $$1.22\lambda/D$$
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

 

[DrDu]

I suppose you have to derive first the size of the Airy disk in the focal plane. The rest is geometry.

 

[eoghan]

I tried, but I'm not sure. The distance II' is
$$II'=f\sin(\theta)$$
where f is the distance of I from the center of the lens.
The images are resolved if
$$II'\geq f\frac{1.22\lambda}{D}$$
If we call f' the distance between O and the center of the lens, we have
$$\frac{OO'}{f'}=\frac{II'}{f}$$
therefore at the limit of the resolution
$$\frac{OO'}{f'}=\frac{f\frac{1.22\lambda}{D}}{f}$$
But from the geometry we know that
$$f'=\frac{\frac{D}{2}}{\tan(i)}\simeq\frac{D}{2sin(i)}$$
Thus, in the end
$$OO'=\frac{1.22\lambda}{2\sin(i)}$$

Is this all right? My doubt is that this derivation is much simpler than that given in textbook...

 

[Andy Resnick]

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).

 

[PJC01]

described above

Hello,

The resolution limit for a microscope is known as the Airy disc, which is the result of diffraction caused by the lens acting as a circular aperture. This means that a point source of light is diffracted into a disk of angular aperture 1.22λ/D, where λ is the wavelength of light and D is the diameter of the lens.

To understand how this limit is obtained, we can look at the work of Jenkins and White. They start by considering two point sources, O and O', with O being on the axis of the lens and O' slightly above it. These two sources form images, I and I', respectively, which consist of disks. The angular separation of these disks when they are on the limit of resolution is 1.22λ/D.

When this condition is met, the wave from O' diffracted to I has zero intensity, meaning that the light is focused to a point. The extreme rays, O'BI and O'AI, differ in path by 1.22λ. This is because the path difference is equal to the difference in distances traveled by the two rays from the source to the lens and then to the image point. Since the light from O' is focused to a point, the path difference can be calculated using basic geometry and is found to be 1.22λ.

I hope this helps to clarify the concept of the Airy disc and how it relates to the resolution limit of a microscope.