Achromatic prisms and lenses
Dispersion
When light passes through a prism the
amount of deviation depends on the
refractive index, and since the refractive
index is different for different wavelengths
the deviation differs for different colours of
light.
If a beam of white light is shone on a prism
as shown in Figure 1 the refracted beam is
separated into a spectrum (for the present
we will restrict ourselves to a consideration
of the visible spectrum).
A
dR dB
red

white light
blue
Figure 1
This spreading of the beam is called dispersion and can be shown to depend both on the
refracting angle of the prism and on the refractive index of the material of which it is made.
If nR and nB are the refractive indices for red and blue light at the extreme ends of the visible
spectrum, then the deviations for red and blue light are:
dR = (nR - 1)A and
dB = (nB - 1)A respectively.
Therefore for a prism of small angle the angular dispersion () is given by the formula:
Dispersion () = dR - dB = (nB - nR)A
The mean deviation for a prism is taken as being that produced with yellow light and is given
by:
Mean deviation (dY) = (nY - 1)A
where nY is the refractive index of the glass of the prism for yellow light.
‘Blue’, ‘red’ and ‘yellow’ are rather vague terms, however, since each colour represents a
range of wavelengths and so for accurate work we choose one particular wavelength within
each area of the spectrum:
for red, the C line of hydrogen with a wavelength of 656 nm
for yellow, the D line of sodium with a wavelength of 589 nm
for blue, the F line of hydrogen with a wavelength of 486 nm
The refractive indices of two types of glass for these three standard wavelengths are given in
the table below:
Crown glass
Flint glass
nC
1.5150
1.6434
nD
1.5175
1.6550
nF
1.5233
1.6648
The accurate definition for mean deviation therefore becomes:
Mean deviation (dD) = (nD - 1)A
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Example
Calculate the angular dispersion produced by a flint glass prism of refracting angle 20o. (Take the
refractive indices for red and blue right to be as shown in the table above.)
Angular dispersion = (1.6648 - 1.6434) x 20 = 0.428o
Dispersive power
A useful property to consider when calculating the dispersion is the dispersive power of a
material. This depends only on the type of material of which a prism or lens is made and not
on its shape. Dispersive power is defined as:
Dispersive power ( = angular dispersion/mean deviation = (nF – nC)/(nD - 1)
Achromatic prisms and lenses
Although the dispersion of white light is useful when we want to look at the spectrum of the
light it is a real problem in optical instruments such as telescopes. The lenses in these
instruments disperse different colours by different amounts and so bring the different colours
to different foci. The images formed are coloured and blurred. It is therefore necessary to
deviate the light without dispersing it, and prisms and lenses that do this are called
achromatic (Greek, ‘without colour’).
(a) The achromatic prism
Such a prism is a compound prism made of two prisms of materials with different refractive
indices, say n and n'.
The dispersion for prism 1 will be: dR - dB = (nB - nR)A
and that for prism 2:
dR' - dB' = (nB' - nR')A'.
For there to be zero dispersion the algebraic sum of these two
dispersions must be zero, and therefore:
(dR - dB) + (dR' - dB') = (nB - nR)A + (nB' - nR')A' = 0
flint
Therefore:
A
The negative sign indicates that the prisms must be placed as
shown in Figure 2
A single ray of white light passing through an achromatic prism
will give rise to a parallel beam of light which when brought to a
focus will appear white again. If we take more than one incident
ray then the colours will overlap, giving a white centre with
coloured edges.
A'
crown
Figure 2
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Example
A crown glass prism of refracting angle 6o is combined with a flint glass prism to give an achromatic
combination. Calculate the refracting angle of the flint glass prism. What deviation will the compound
prism produce? (Take the refractive indices to be those in the table above.)
Let A be the angle of the flint glass prism. Then:
A/6 = - [ 1.523 - 1.515]/[1.665 - 1.643]
giving A = - 2.2o
Deviation of red light = (1.515 - 1) x 6 - (1.643 - 1) x 2.2 = 1.68o.
(b) The achromatic lens
The dispersion of lenses can be a serious problem in large astronomical instruments - for
example, the difference in focal length for red and blue light for a telescope with a mean focal
length of around 15 m can be as much as 45 cm. (An exaggerated version of the defect is
shown in Figure 3). Such a difference is obviously quite unacceptable when a clearly focused
image is required.
White light
FB
FR
Figure 3
This defect of lenses is known as chromatic aberration.
For a lens to be achromatic the focal length for red light (FR) must be the same as that for
blue light (FB). As with the achromatic prism this can be produced by using a ‘doublet’ made
of two thin lenses of different refractive indices (Figure 4).
For blue light: 1/FB = 1/fB + 1/fB'
For red light: 1/FR = 1/fR + 1/fR'
and also we have for each lens:
1/fB - 1/fR = (nB – nR)(1/R1 + 1/R2)
and
1/fy = (nY – 1)(1/R1 + 1/R2)
n1
Therefore:
1/fB - 1/fR = /fY and 1/fB' – 1/fR' = '/fY'
Therefore:
This gives:
/fY + '/fY' = 0
/fY + '/fY' = 0
In this formula the negative sign means that one of the lenses is convex and
the other concave.
Figure 4
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Notice that we have only made the lens truly achromatic for two colours, red and blue. There
will still be a spread of colour due to the other wavelengths.
It is possible to make an achromatic lens using two thin lenses of the same material if they
are separated by a distance equal to the mean of their focal lengths.
Defects of lenses
In addition to chromatic aberration described above, lenses suffer from several other defects.
(a) Spherical aberration
This is a result of the inner and outer portions of a lens having different focal lengths, that of
the outside being shorter than that of the centre.
One way of reducing this is to make the deviation at the two surfaces as nearly equal as
possible. Spherical aberration is therefore particularly marked when using a piano-convex
lens with parallel light hitting the plane face.
Spherical aberration is also reduced by decreasing the aperture of a lens and by increasing
its focal length.
(b) Coma
This defect produces a comet-like tail added to all images. It results from off-axis objects
coupled with the different magnifications of different zones of the lens.
The rays from the vertical plane intersect in a horizontal line while those from a horizontal
plane intersect in a vertical line.
(c) Astigmatism
If the object point lies off the axis of the lens then the rays from the horizontal and vertical
planes come to a focus at different distances from the lens.
(d) Distortion
The magnification of the lens varies from its centre to its edge and so the magnification of the
image will vary as well. This gives rise to distortion.
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