|
Resolving power (Page 2)
Resolving power, or resolution, is the smallest distance between two
separate points of an object, when viewed with an optical instrument, that can
still be seen as distinguishable. A microscope's resolution limit, d, can be
found by the following formula:
d = 0.61 λ
/ NA, where λ
is the wavelength of light coming from the object, and NA is the numerical
aperture. (This is called the Ernst Abbe formula)
|
"Proof" of d = 0.61 λ
/ NA:
let:
-
λ
= wavelength
- u = angle of the cone of light coming from object
- u' = angle of cone of light forming image
- n = refraction index of object
- m = magnification
- NA = numerical aperture
- d = distance between two points in the image
d = 0.61 λ
/ (m tanu') (1)* **
m = n sinu / sinu', thus m sinu' = n sinu = NA
However, tanu is approximately sinu when u is very small.
Therefore, m tanu' = NA (2)
Substituting (2) into (1): d = 0.61 λ
/ NA
*this formula is derived in "Theory of Optical instruments"
** all = not strict equalities but approximations
Note: A detailed proof
of this formula can be found in "Theory of Optical Instruments"pp53-54, 69
& "Geometerical Optics" pp99-100, 125. However, to understand these proofs
a high level of Mathematics is required
|
The resolving power increases when d, the minimum distance that can be seen
between two points in the image, decreases. Thus, according to the formula d =
0.61 λ
/ NA, the resolving power can be increased in two ways:
- decreasing the wavelength, λ
(ie by using filters)
- increasing the NA. As stated earlier, NA = n sinu. Thus, NA can be
increased the following ways:
- increasing the refraction index, n (this can be done by adding oil to
the object)
- increasing the angle of light coming from the object, u
The limiting angle of resolution can be decreased when:
- the diameter of the lens, b, is increased
- decreasing the wavelength, λ
(ie by using filteres)
Note: Theoretically, the largest value of u should be 90 degrees (assuming the
lens was large enough); however, in practice, the maximum view is only about
71.8 degrees (ie sinu = 0.95). Even though 71.8 degrees is the maximum angle
that can be obtained, it is every difficult to achieve such a high u since
equipment and environment must be ideal.
|
What is the limit of resolving power using a light
microscope?
Let NA = 1.4.
The resolving power depends on the colour (or wavelength) of light. If
looking at green light (the colour eyes are most sensitive to), λ
= 500nm, thus:
r = 0.61 x 500nm / 1.4 = 218nm
If using blue light (which has the smallest wavelength), resolution is:
r = 0.61 x 400nm / 1.4 = 174nm
If using red light (which has the largest wavelength), resolution is:
r = 0.61 x 700nm / 1.4 = 305nm
So, the range of "best" resolution is about 200nm to 300nm |
Notes:
- the wavelength of visible light ranges from 400-700nm
- A distance of 1/3 λ
between two objects will have no resolution.
- An electron microscope has a better resolution since electrons have a
shorter wavelength than light. In fact, the highest resolving power of an
electron microscope is about 0.1nm. This is about 1000 times better than that
of a light microscope!
- An increase in magnification does not help you see finer details when
resolution is at its max since there is no detail to be seen. In fact,
brightness decreases when the magnification increases so a greater
magnification may hinder the resolution of the image.
|