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. - Physics (Theory)

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.

Solution

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 fr = fv 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 Fv and Fr 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
Fv = Fr
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.

Concept: Lenses
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