Resolution and Pupil size
Is the resolving power of the eye affected by the
size of the pupil?
Introduction
Resolution is the ability to see fine detail in an image; it can be
quantified by separation of two distinguishable points. According to
text books at this level (Giancoli, 1995) the resolution an optical
instrument is dependent on the size of the aperture; this is one of
the reasons why high resolution telescopes have such large diameter
mirrors (Hamper, 2009). The eye is an optical instrument that has a
small aperture (the pupil) according to the same principle its
resolution should therefore be dependent on the pupil size which
would imply that our eyes could resolve more detail in the dark
when the pupil is large than in bright light when it is small, this
seems strange so in this essay the question “Is the resolving power
of the eye affected by the size of the pupil?” will be addressed.
Diffraction by small apertures
When light passes through a small aperture it spreads out to
produce an intensity pattern shown in fig.1, this is called diffraction
and can be explained using Huygens’ construction. This models the
wave front as an infinite number of small wavelet sources each
propagating a spherical wave in the forward direction, these
wavelets can be summed to give the resultant intensity at any given
angle. Using this model it can be shown that the angular separation
of the first minima θ = 1.22λ/a where a is the diameter of the
aperture. If an image of a point object is produced by an optical
instrument diffraction at the aperture will cause the image to be a
series of rings rather than a point.
Fig. 1 Diffraction of light by a circular
aperture
Rayleigh Criterion
The Rayleigh criterion is a way of defining the resolving power of an
optical instrument, this states that “two images are just resolvable
when the centre of the diffraction disk of one is directly over the first
minimum of the other” (Giancoli, 1995 p.727). This is illustrated in
fig.2. The outcome of this is that the resolving power of an optical
instrument is dependent on the size of the aperture.
Fig. 2 Rayleigh Criterion
The Human Eye
Light enters the eye (fig.3) through the pupil and is focused on the
retina by the lens. The retina is made up of millions of light sensitive
cells that convert the light to a nerve impulse that can be sent to the
brain via the optic nerve. The cells of the retina do not work well if
the light is either too bright or too dark; to optimise the amount of
light entering the eye the iris controls the size of the pupil making it
smaller when the light is bright and bigger when the light is dark.
The average size of the pupil can vary from 3-9mm (Wikepedia,
2010). Since the amount of light entering the eye is proportional to
the diameter squared the pupil can vary the light by a factor of 9,
this is not enough to deal with the difference from seeing in bright
sunlight to seeing stars at night a factor 107 (Clark R.N. 2005), this is
managed by chemical changes that take place in light sensitive cells
themselves combined with using different cells for low light
conditions.
The resolving power of the eye
The resolving power can be quantified by the angle subtended
between the two closest points that can be seen apart.
Using the Rayleigh criterion it is possible to calculate a value for the
resolving power of the eye.
If the wavelength of light is 5nm and the diameter of the pupil is
5mm the using the equation θ = 1.22λ/a we can deduce that the
points will not be resolved if the angle between them is less than
1.2x 10-4 radians. This means that if two objects 1mm apart will be
able to be distinguished at a distance of 1/1.2x10-4 mm,
approximately 8m.
Fig. 4 Angle subtended by two point objects
Fig. 3 The Human eye
Measuring the Resolving Power of the Eye
To measure the resolving power of the eye two close together
objects are viewed close to the eye, the distance between the eye
and the objects is then increased until they can no longer be seen as
two separate lines. A commonly used variation of this (Stokes H.T.) is
to use a series of lines and decide at what distance they appear to be
continuous. Fig.5 shows an example of this. As the page is moved
from the eye, at some distance x the lines appear to be the same as
the filled square. If the separation of the lines is d then the resolving
power is d/x radians.
A trial of this with a line separation of 0.78mm revealed that the
lines became indistinguishable at a distance of 185 ±2 cm this gives a
resolving power of about 4 x 10-4 rads which is not as good as that
predicted by the Rayleigh criterion.
Measuring the size of the pupil
Since the experiment was to be performed alone the easiest way to
measure the pupil size was by using a camera. This was set up at a
fixed distance from the eye and a picture recorded. The picture was
then uploaded to my computer and the programme LoggerPro
(Vernier) was used to measure the pupil size. To scale the
measurements a mm scale was placed close to the eye. Fig.6 is an
example of one of the photographs.
Fig. 6 Measuring the eye
In this photograph the scale was set using the ruler on the left, the
eye was then measured at several places and found to have an
average diameter of 0.49 ± 0.03cm. To make the pupil change size
the light intensity had to be changed, this caused problems when
photographing the eye, it was also very difficult to carry out the
resolution test and measure the eye at the same time for this reason
it was decided not to measure the eye pupil but to carry out the test
at different light intensity with the assumption that the pupil size will
increase with decreasing light intensity.
Fig. 5 The resolving power test
Measuring resolving Power with different pupil sizes
To test the resolving power of the eye a grid similar to fig.5 was
printed onto white paper and mounted on a wall. To change the
pupil size the light intensity needed to be varied but doing that
would make the room dark therefore making the lines, more difficult
to see so the challenge was to increase the size of the pupil without
making the room dark. The solution used was to place a black tube in
front of the eye; this would reduce the amount of light incident on
the eye without making the object appear dark. By increasing the
length of the tube the light falling on the eye would become less and
the pupil bigger.
The way this works can be illustrated in fig.7. Where the observer is
viewing light from an illuminated wall, light at a wider angle than
that shown cannot pass down the tube into the eye so as the tube
length is increased so the amount of light entering the eye is
reduced.
Fig. 7 varying light intensity with a tube
To test the effectiveness of this system a light sensor was placed at
the end of the tube as its length was increased; the results are
shown in graph 1.
Preliminary trials of the experiment highlighted a problem regarding
my eyesight. I am longsighted which means I have to wear glasses to
read, without glasses I cannot focus on objects closer than about 3m
and beyond that I can’t focus with glasses. For that reason The grid
used was big enough to be resolved from a great enough distance so
that I did not have to use glasses.