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).

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 n_R and n_B 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:

d_R = (n_R - 1)A and d_B = (n_B - 1)A respectively.

Therefore for a prism of small angle the angular dispersion (f) is given by the formula:



The mean deviation for a prism is taken as being that produced with yellow light and is given by:


where n_Y 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:


The accurate definition for mean deviation therefore becomes:

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:

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: d_R - d_B = (n_B - n_R)A
and that for prism 2: d_R' - d_B' = (n_B' - n_R')A'.

For there to be zero dispersion the algebraic sum of these two dispersions must be zero, and therefore:
(d_R - d_B) + (d_R' - d_B') = (n_B - n_R)A + (n_B' - n_R')A' = 0

Therefore:

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.





(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.




This defect of lenses is known as chromatic aberration.

For a lens to be achromatic the focal length for red light (F_R) must be the same as that for blue light (F_B). 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/F_B = 1/f_B + 1/f_B' For red light: 1/F_R = 1/f_R + 1/f_R'

and also we have for each lens:

1/f_B - 1/f_R = (n_B – n_R)(1/R₁ + 1/R₂) and 1/f_y = (n_Y – 1)(1/R₁ + 1/R₂)

Therefore:

1/f_B - 1/f_R = w/f_Y and 1/f_B' – 1/f_R' = w'/f_Y' This gives: w/f_Y + w'/f_Y' = 0


Therefore:

In this formula the negative sign means that one of the lenses is convex and the other concave.
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.