The limit of resolution
(or resolving power) is a measure of the ability of
the objective lens to separate in the image adjacent details
that are present in the object. It is the distance between
two points in the object that are just resolved in the
image. The resolving power of an optical system is
ultimately limited by diffraction by the aperture. Thus an
optical system cannot form a perfect image of a point.
For resolution to occur, at
least the direct beam and the first-order diffracted beam
must be collected by the objective. If the lens aperture is
too small, only the direct beam is collected and the
resolution is lost.
Consider a grating of
spacing d illuminated by light of wavelength
l, at an angle of
incidence i.
The path difference between
the direct beam and the first-order diffracted beam is
exactly one wavelength, l.
So,
d
sin i + d sin a
= l
where 2a
is the angle through which the first-order beam is
diffracted. Since the two beams are just collected by the
objective, i =
a, thus the limit of
resolution is,
The wavelength of light is
an important factor in the resolution of a microscope.
Shorter wavelengths yield higher resolution. The greatest
resolving power in optical microscopy requires
near-ultraviolet light, the shortest effective visible
imaging wavelength.
Numerical Aperture
The numerical aperture of a
microscope objective is a measure of its ability to resolve
fine specimen detail. The value for the numerical aperture
is given by,
Numerical
Aperture (NA) = n sin
a
where n is the
refractive index and equal to 1 for air and
a is the half angle
subtended by rays entering the objective lens.
Numerical aperture
determines the resolving power of an objective, the higher
the numerical aperture of the system, the better the
resolution.
Low
numerical aperture
Low value for a
Low resolution |
High numerical aperture
High value for a
High resolution |
Airy Discs
When light from the various
points of a specimen passes through the objective and an
image is created, the various points in the specimen appear
as small patterns in the image. These are known as Airy
discs. The phenomenon is caused by diffraction of light as
it passes through the circular aperture of the objective.
Airy discs consist of
small, concentric light and dark circles. The smaller the
Airy discs projected by an objective in forming the image,
the more detail of the specimen is discernible. Objective
lenses of higher numerical aperture are capable of producing
smaller Airy discs, and therefore can distinguish finer
detail in the specimen.
The limit at which two Airy
discs can be resolved into separate entities is often called
the Rayleigh criterion. This is when the first diffraction
minimum of the image of one source point coincides with the
maximum of another.
|
|
|
Unresolvable |
Rayleigh Criterion |
Resolvable |
Circular apertures produce
diffraction patterns with circular symmetry. Mathematical
analysis gives the equation,
qR
is the angular position of the first order diffraction
minimum (the first dark ring)
l is the wavelength of
the incident light
d is the diameter of the aperture
From the equation it can be
seen that the radius of the central maximum is directly
proportional to l/d.
So, the maximum is more spread out for longer wavelengths
and/or smaller apertures.
The primary minimum sets a
limit to the useful magnification of the objective lens. A
point source of light produced by the lens is always seen as
a central spot, and second and higher order maxima, which is
only avoided if the lens is of infinite diameter. Two
objects separated by a distance less than
qR cannot be
resolved.