Resolving Power: Abbe vs. Rayleigh Criterion

[KFC]

Hi there,
I am reading some material on resolving power of lens and diffraction effect. As I known, the first on who consider the relation of diffraction and resolution is E. Abbe in 1873, who gave the following relation

\sin\alpha = \lambda / (2 n D)

where n is the index of refracion and D is aperature diameter. However, in the text of optics, I found something similar but different

\sin\alpha = 1.22 \lambda / D

so what's the difference between these? How does the 1.22 come from?

BTW, later in the text, I also read a criterion call Rayleigh's criterion which just approximate \sin\alpha as \alpha (I guess), so does Rayleigh's criterion only an approximation of Abbe's expression?

 

[Lok]

Maybe they use a standard value for n and don't bother further. n = 1.6... reasonable lens material.

 

[Lok]

I forgot ... does the n stand for the refreactive index of the lens or the medium or both?

 

[Andy Resnick]

Hi there,
I am reading some material on resolving power of lens and diffraction effect. As I known, the first on who consider the relation of diffraction and resolution is E. Abbe in 1873, who gave the following relation

\sin\alpha = \lambda / (2 n D)

where n is the index of refracion and D is aperature diameter. However, in the text of optics, I found something similar but different

\sin\alpha = 1.22 \lambda / D

so what's the difference between these? How does the 1.22 come from?

BTW, later in the text, I also read a criterion call Rayleigh's criterion which just approximate \sin\alpha as \alpha (I guess), so does Rayleigh's criterion only an approximation of Abbe's expression?

The second formula relates to the minimum size of an imaged point, and is called the Rayleigh criterion. That is, a point object will image to an Airy disk (insert caveats here), and the factor 1.22 is the full-width half-max of the Airy function (or sombrero function, or J_0(ax)/ax). This means two points have to be separated by a certain distance to be resolved as two points. The Rayleigh criterion was derived based on telescopes observing distant stars.

The first formula looks like the Abbe criterion, and is also related to the minimum resolving power of a lens. There are some slight nuances between the two (the Abbe criteria was derived based on Bragg scattering), but the bottom line to remember is that "resolving power" is not well-defined in general.

The approximation sin(a) ~ a is not Rayleigh's criteria, it's the paraxial approximation, and is used in geometrical optics.

Does that help?

 

[mgb_phys]

It's also worth knowing that the Airy criterion (the 1.22) is only an arbitrary limit picked by Airy - it's roughly the point at which you can distinguish two stars by eye. There is information in the image below this limit

The n is the general case but it is of the medium between the lens and the object which is almost always either space (n=1) or air (n=1 and a bit) so it gets forgotten about.

 

[KFC]

Thank you all of you. Now it is clear.

 

[KFC]

It's also worth knowing that the Airy criterion (the 1.22) is only an arbitrary limit picked by Airy - it's roughly the point at which you can distinguish two stars by eye. There is information in the image below this limit

The n is the general case but it is of the medium between the lens and the object which is almost always either space (n=1) or air (n=1 and a bit) so it gets forgotten about.

So ... you call the criterion (the 1.22) as Airy criterion? I wonder who, Airy or Rayleigh, is the first one who propose that criterion? Do you know which paper first present this idea?

 

[mgb_phys]

Sorry should be the Raleigh criterion (another British astronomer around the same time) the distribution of the light is an Airy function (invented by Airy before Raleigh was born) but the limit is due to Raleigh