Derive an expression for the focal length of the combination of two thin lenses when they are SEPERATED by a small distance?
Consider two lenses L1 and L2 separated by a small distance 'd' apart, as shown below. A ray of light AB initially parallel to the principal axis hits the lens L1 and deviates and then hits lens L2.
f1 is the focal length of L1 and
f2 is the focal length of L2
δ1 is the deviation produced by L1
δ2 is the deviation produced by L2
form simple geometry
δ1 = h1 / F1
δ2 = h2 / F2
total deviation of the ligth ray will be
δ = δ1 +δ2
δ = (h1 / F1) + (h2 / F2)
at lens 2 and triangle BCD
h2 = h1 - CD = h1 - BD.tanδ1
h2 = h1 - d.tanδ1
now as δ1 is very small, tanδ1 ~ δ1
h2 = h1 - d.δ1
so, from earlier relation
h2 = h1 - d.(h1/f1)
δ = δ1 +δ2 = (h1/f1) + [(h1 - d.(h1/f1)) / f2]
δ = (h1/f1) + (h1/f2) - (dh1 / f1.f2)
for the combination of the two lenses let F be the combined focal length.
So, the total deviation will be given as
δ = h1 / F
(h1/f1) + (h1/f2) - (d.h1 / f1.f2) = h1 / F
1/F = 1/f1 + 1/f2 - (d / f1.f2)
so, the combined focal length will be
F = f1.f2 / (f1 + f2 - d)
Combination of Thin Lenses in Contact
Consider two lenses A and B of focal length f1and f2placed in contact with each other. An object is placed at a point O beyond the focus of the first lens A. The first lens produces an image at I1(real image), which serves as a virtual object for the second lens B, producing the final image atI.
Since the lenses are thin, we assume the optical centres (P) of the lenses to be co-incident.
For the image formed by the first lens A, we obtain
For the image formed by the second lens B, we obtain
Adding equations (i) and (ii), we obtain
If the two lens system is regarded as equivalent to a single lens of focal length f, we have
From equations (iii) and (iv), we obtain
For several thin lenses of focal length f1, f2, f3, …, the effective focal length
In terms of power, equation (vi) can be written as