Lenses and Ray Diagrams

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Lenses and Ray Diagrams

Both converging and diverging lenses have a principal axis through the
centre of the lens and perpendicular to it. The principal focus F (aka focal
point) lies on the principal axis, at the focal distance (focal length).
Rays from a distant object travel parallel to the principal axis and are
refracted through F. (In the case of a diverging lens, they appear to have
come from F on the same side as the object).

Constructing ray diagrams

There are three rays required to construct a ray diagram for a lens (see
Ordinary Level Physics by A F Abbott, p272-273 in my edition).

(1)   Rays parallel to the principal axis are refracted through the principal
focus F.
(2) Rays through the principal focus F are refracted to emerge
parallel to the principal axis.
(3) Rays through the optical centre of the lens do not change
course.

2F           F                        F            2F     principal
axis

lens

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Diverging lens

Object      F   Image   F    principal
axis

Construction

Object      F   Image   F   principal
axis

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Converging lens
There are a number of different cases here.

1) Object at Infinity

2F             F                     F   2F   principal
axis

lens

2) Object beyond 2F

Object   2F             F                     F   2F   principal
axis

lens

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3) Object at 2F

Object         F               F   2F   principal
2F                                    axis

lens

4) Object between F and 2F

2F Object   F               F   2F   principal
axis

lens

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5) Object at F

2F       Object             F   2F   principal
F                         axis

lens

6) Object between lens and F

2F         F      Object    F   2F   principal
axis

lens

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Focusing power
The greater the curvature of the lens, the greater its strength or power.
Obviously it also reduces its focal length at the same time (by implication
then, a short focal length means a powerful lens).

So        power = 1/focal length (in metres)                Units: dioptres (D)

We say converging lenses have a positive power, while diverging lenses
have a negative power.

The human eye
The cornea itself has a power of about +43D. The eye lens varies from
about 17D at its thinnest (unaccommodated) state, to about +31D at its
thickest (accommodated) state. So the eye’s total power ranges from
Even so, there is a limit to how close an object can be focused. This is
called the near point, and it gets further away with age – this is called
presbyopia, and is the reason why older people tend to hold their
newspapers at arm’s length until they succumb to wearing reading
glasses!

The eye’s sensitivity to light

to nerve cells            The retina is made up of two types of
cells, rods and cones.
Rods are sensitive to low light levels.
Cones are colour sensitive; the three
nucleus                 types respond to the colours red, green
and blue. They are less sensitive than
rod                           cone   rods, especially at very low and very high
cell                          cell   light intensities.

Both types use a chemical mechanism to
detect light. Once a cell has been
triggered, it cannot detect light until the
molecules have regenerated. Up to ten photons may be needed to trigger
a rod cell. It can take 30 minutes for the eyes to adapt from bright sunshine
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to very dim light, due to the need for the molecules to regenerate. (note
figure 5, p25, CAMS SSS)
The response of the eye depends on intensity, but is non-linear. (fig 24,
p95, CAMS SSS) shows this; the curve for the cones is a logarithmic one.

Response  log(intensity)

(The details of the chemical mechanism are not required)

Colour vision and persistence

The different cell types are sensitive to different wavelengths of light.
rods
relative
(think: what is the
sensitivity                                                            wavelength of the sodium
light used in street lamps?)
green

In low light, blue-green
red                           objects are more easily seen
blue                                                     than red. But little colour is
seen at all, because the rod
cells are used
400   500        600          700   wavelength (nm)
predominantly.

A bright image will cause the cones of that colour to trigger, and will be
unable to re-trigger for a while. In the meantime, the other ones operate
normally to form an after-image in the complementary colour. This is an
example of persistence of vision. Cinematic cameras rely on this; although
they only show 24 frames per second, the eye records a smooth image.

Resolution

This is determined by the cells of the retina rather than by the physical
optics. The average spacing of retinal cells is 3m. However the central
area of the retina - the fovea – has no rods, only tightly packed cones.
This part allows a visual angle (measured from the optical centre of the
eye) of about 0.5 minutes or 1/120 degree.
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Quantum Efficiency
This is the number of nerve fibres triggered as a percentage of the number
of photons entering the eye. Up to 10 photons may be needed to trigger
one cell, and several cells must be stimulated to trigger a single nerve
fibre. Combined with a quantity of the photons being absorbed by the
vitreous and aqueous humours, the overall quantum efficiency is only

The Camera

The camera mimics the eye in many ways. A converging lens focuses the
light, but is focused by moving it closer to of further from the film on a
The shutter prevents light from entering until required (like the eyelid). The
speed or time interval of the shutter controls the amount of light allowed to
enter (and hence the exposure time).
The aperture or iris controls the amount of light entering in any given time
interval.
The film detects the image, like the retina, but stores it permanently.

Apertures and f-numbers
An exposure is a controlled combination of shutter speed and aperture (f-
stop). Fast speeds are used for action pictures, small apertures give a
greater range of focus.
The f scale is exponential, in that each value has half or twice the aperture
area of the next. It is decided by

Relative area = focal length
f-number

f-number     1    2      2.8   4      5.6        8      11      16      22
Aperture     F    F/2    F/2.8 F/4    F/5.6      F/8    F/11    F/16    F/22
width
Relative     1    ¼      1/7.5 1/16 1/31.4       1/64   1/128 1/256 1/484
aperture
area
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Depth of field
…is the range of object distances which can be clearly focused on the film.
Small apertures give greater depth of field.

Depth of focus
…is the range of image distances which can be clearly focused on the film.

diffracted
O
O2                                                             I

aperture
stop

Photographic films

A film is made up of millions of tiny silver halide (usually bromide) crystals
suspended in gelatin. The crystals are referred to as grains, and can be
between 3m and 20nm across according to film type. Larger grains are
more sensitive to light (“high-speed” films, eg 1000ASA).
Each grain needs about 10 photons to “expose” it. Only about 1 in 10
photons are absorbed by grains, so about 100 photons are required to
expose each grain (a similar quantum efficiency to the eye).
When a grain is exposed, the silver bromide changes to leave a small
particle of silver. The film darkens as it is exposed, so light areas of the
image are represented by dark areas on the film, ie “negative”.
Faint or dim images can be improved by increasing the time the film is
exposed for. In a longer time period, more photons are received by the
film, which “integrates” the exposure effect. This is particularly applicable
when photographing relatively static and very faint images such as stars.

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