2.12: Principal Planes
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Consider a thick lens, or a system of two separated lenses. In Figure II.21, F1 is the first focal point and H1 is the first principal plane. In Digure II.22, F2 is the second focal point and H2 is the second principal plane
I refer now to the second part of FFigure II.22, and I suppose that the focal lengths of the two lenses are f1 and f2, and the distance between them is D. I now invite the reader to calculate the distances x2 and y2. The distance x2 can be calculated by consideration of some similar triangles (which the reader will have to add to the drawing), and the distance y2 can be calculated by calculating the convergences C1,C2,C3,C4 in the manner which is by now familiar. You should get
x2=Df2f1+f2−D.
and
y2=f2(f1−D)f1+f2−D.
I further invite the reader to imagine that the two lenses are to be replaced by a single lens situated in the plane H2 so as to bring the light to the same focus F2 as was obtained by the two original lenses. The question is: what must be the focal length f of this single lens? The answer is obviously x2+y2, which comes to
f=f1f2f1+f2−D.
The eyepiece of an optical instrument such as a telescope or a microscope is generally a combination of two (or more) lenses, called the field lens and the eye lens. They are generally arranged so that the distance between the two is equal to half the sum of the focal lengths of the two lenses. We shall now see that this arrangement, with two lenses made of the same glass, is relatively free from chromatic aberration.
Let us remind ourselves that the power of a lens in air is given by
P=1f=(n−1)(1r1+1r2)
Here r1 and r2 are the radii of curvature of the two surfaces, and n is the refractive index of the glass. For short, I am going to write Equation ??? as
P=1f=μS,
where μ=n−1 and S=(1r1+1r2). That being so, Equation ??? can be written
P=μ(S1+S2)−μ2SD1S2
This equation shows how the position of the focus F2 varies with colour. In particular,
dPdμ=S1+S2−2μDS1S2,
which shows that the position of F2 doesn’t vary with colour provided that the distance between the lenses is
D=S1+S22μS1S2.
On going back to Equation ???, we see that this translates to
D=12(f1+f2).__