Imaging with a thin lens
5.1.2 Imaging with a thin lens
Let us now consider what are called thin lenses using as an example a biconvex lens made from crown glass. It is not possible to prove that there is constant path length from a plane wavefront perpendicular to the optical axis to the focus as it is not true but, for thin lenses, it is very nearly true.
If the focus (where the wavelets add coherently) is the point where all path lengths from a plane wavefront passing through the lens have taken the same time to traverse, then it must be true for the two paths (ray 1) through the centre of the lens and (ray 2) a path that just passes through the tip of the lens where we will assume the additional delay by passing through the glass is zero (Figure 5.5).
Observing the Universe
Figure 5.5 The geometry of imaging with a biconvex lens.
You can immediately see that a lens works by delaying the light passing through its centre by just the same amount as the delay caused by the extra path length in space that the edge ray has to traverse. The delay is due to the fact that the effective path length through glass is increased over that in a vacuum by a factor which is given by the refractive index of the glass, n. This varies with different glass types and also, of extreme importance, with the wavelength of the light passing through it – a phenomenon called dispersion. The refractive index of crown glass at the wavelength of green light is typically 1.5. (Air essentially acts like a vacuum – it scarcely delays light having a refractive index of just 1.0008 so we will assume that the thin lens is in a vacuum.)
Simple geometry equating the time it takes to traverse the two extreme paths and calculating the thickness of the lens in relation to its diameter and radius of curvature derives a formula giving the focal length as a function of the radius of curvature of the lens and the refractive index of the glass. The resulting equation
is exactly that derived by ray optics and is the simplifi ed form of the lensmaker’s
equation when the radii of curvature of the two lens surfaces are the same. From Figure 5.5:
(f 2 2 ) 1/2
(f 2 2 ) 1/2 (f 2 2 ) 1/2
Square both sides: f 2 2 ⫽f 2 2
We can ignore the fi nal term. For a thin lens t is very much less than f so the squared term in t is far smaller that the two other parts of the right-hand side of the equation.
Introduction to Astronomy and Cosmology
In addition we can cancel out the f 2 term giving: r 2 ⫽ 4ft(n ⫺ 1)
f ⫽r 2 /4t(n ⫺ 1)
If the radius of curvature of the lens surface is a, then:
a 2 ⫽ (a ⫺ t) 2 2
Expanding the right-hand side gives:
a 2 ⫽a 2 2 2
We can again ignore the t 2 terms and cancelling out the a 2 terms gives:
2at ⫽ r 2
That is:
a ⫽r 2 /2t Substituting for r 2 / 2t in the expression for f gives: f ⫽ a/2(n ⫺ 1)
This is the lensmaker’s equation for a thin biconvex lens whose surfaces have the same radii of curvature.
Table 5.1 gives the refractive indices of both a crown and a fl int glass for three colours of light. We will fi rst consider a biconvex lens made of crown glass, and choose a radius of curvature, 1000 mm, which gives a focal length typical of an astronomical refracting telescope.
f blue ⫽ 954 mm f green-yellow ⫽ 967 mm f red ⫽ 970 mm
Observing the Universe
Table 5.1 The refractive indices of crown and fl int glass at three wavelengths. Blue
Red 486.1 nm
It is immediately apparent that the single lens gives red and blue focuses that are widely separate from the green-yellow focus. Using such a lens in a telescope and trying to get the best image – probably where the green light comes to a focus – one would see the sharp green image superimposed on out of focus blue and red images. Blue and red together make purple, so the in focus image is surrounded by a purple glow. This effect is called chromatic aberration. A small diameter lens with a relatively long focal length can give
a passable image and we should not forget that Galileo was able to show that the planets orbit the Sun by making some superb observations of Venus with just such a telescope.