Resolving Power: Abbe vs. Rayleigh Criterion

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In summary, the conversation discusses different criteria for determining the resolving power of a lens and the diffraction effect. The first formula is known as the Abbe criterion, while the second formula is referred to as the Rayleigh criterion. The 1.22 factor in the Rayleigh criterion is based on the full-width half-max of the Airy function. The conversation also mentions the paraxial approximation and the role of the refractive index in determining resolving power. Both the Airy and Rayleigh criteria were proposed by British astronomers in the 19th century.
  • #1
KFC
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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

[tex]\sin\alpha = \lambda / (2 n D)[/tex]

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

[tex]\sin\alpha = 1.22 \lambda / D[/tex]

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 [tex]\sin\alpha[/tex] as [tex]\alpha[/tex] (I guess), so does Rayleigh's criterion only an approximation of Abbe's expression?
 
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  • #2
Maybe they use a standard value for n and don't bother further. n = 1.6... reasonable lens material.
 
  • #3
I forgot ... does the n stand for the refreactive index of the lens or the medium or both?
 
  • #4
KFC said:
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

[tex]\sin\alpha = \lambda / (2 n D)[/tex]

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

[tex]\sin\alpha = 1.22 \lambda / D[/tex]

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 [tex]\sin\alpha[/tex] as [tex]\alpha[/tex] (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?
 
  • #5
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.
 
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  • #6
Thank you all of you. Now it is clear.
 
  • #7
mgb_phys said:
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?
 
  • #8
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
 
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1. What is the difference between Abbe and Rayleigh criteria for resolving power?

The Abbe and Rayleigh criteria are two different methods used to measure the resolving power of an optical instrument. The Abbe criterion takes into account the numerical aperture and wavelength of light, while the Rayleigh criterion considers the size of the diffraction pattern produced by the instrument.

2. Which criterion is more commonly used in modern scientific research?

The Rayleigh criterion is more commonly used in modern scientific research, as it provides a more accurate and precise measurement of resolving power. Additionally, advancements in technology have made it easier to measure and calculate the diffraction pattern size.

3. Can the resolving power of an instrument be improved by adjusting the wavelength of light used?

Yes, the resolving power of an instrument can be improved by using a shorter wavelength of light. This is because shorter wavelengths have a smaller diffraction pattern size, resulting in a higher resolving power according to the Rayleigh criterion.

4. How does the numerical aperture affect the resolving power of an instrument?

The numerical aperture (NA) is a measure of the light-gathering ability of an optical instrument. The higher the NA, the higher the resolving power according to the Abbe criterion. This is because a higher NA allows for more light to enter the instrument, resulting in a sharper image.

5. Are there any limitations to using the Rayleigh criterion for measuring resolving power?

One limitation of the Rayleigh criterion is that it assumes a perfect circular aperture and a point light source. This may not always be the case in real-life instruments, leading to a slight overestimation of the resolving power. Additionally, the Rayleigh criterion does not take into account other factors such as lens aberrations, which can also affect resolving power.

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