Question

 Show that, two thin lenses kept in contact, form an achromatic doublet if they satisfy the condition: `ω/f + (w')/(f') = 0`
where the terms have their usual meaning.

Answer 1

The focal length of a single lens is different for different colors. The image formed by a single lens suffers from chromatic aberration. However, it is possible to combine two lenses of different materials and focal lengths to form an achromatic combination in which f_r = f_v and image is free from chromatic aberration.
According to the lens maker’s formula

`1/f_v  = ( μ_v - 1 )( 1/"R"_1 - 1/"R"_2)`

and `1/f_r  = ( μ_r - 1 )( 1/"R"_1 - 1/"R"_2)`

`1/f_v  -  1/f_r  = ( μ_v - μ_r )( 1/"R"_1 - 1/"R"_2)`   ....(i)

If f is the mean focal length of the lens, then

`1/f = ( μ - 1 )( 1/"R"_1 - 1/"R"_2)` 

or `1/"R"_1 - 1/"R"_2 = 1/(( μ - 1 )f)`

Putting this value in (i), we get

`1/f_v  - 1/f_r  = ( μ_v - μ_r )/(( μ - 1)f) = "ω"/f`    ...(ii)

Similarly, for the second lens of dispersive power ω' and mean focal length f', we write

`1/(f'_v) - 1/(f'_r) = (ω')/(f') `                                ...(iii)

If F_v and F_r are the focal lengths of the combination for violet and red colours respectively, then

`1/f_v  = 1/f_v  + 1/(f'_v)`                               ....(iv)

and `1/F_r = 1/f_r  + 1/(f'_r)`

For an achromatic combination
F_v = F_r
`1/f_v  + 1/(f'_v)  = 1/f_r  + 1/(f'_r)`

`( 1/f_u  - 1/f_r ) + ( 1/(f'_v)  + 1/(f'_r) )` = 0

or `ω/f + (ω')/(f')` = 0 which is the required condition.