In case of two thin lenses in contact
`1/F=1/(f_1)-1/(f_2)`
`=>` `-(dF)/(F^2)=-(df_1)/(f_1^2)-(df_2)/(f_2^2)`
The combination will be free from chromatic aberration if `dF = 0`
i.e., `(df_1)/(f_1^2) + (df_2)/(f_2^2)=0`
which in the light of equation (5) reduces to
`(omega_1f_1)/(f_1^2) + (omega_2f_2)/(f_2^2)`
i.e., `(omega_1)/(f_1) + (omega_2)/(f_2)=0............(6)`
This condition is called condition of achromatism (for two thin lenses in contact) and the lens combination which satisfies this condition achromatic lens. From this condition, i.e., from equation (6) it is clear that in case of achromatic doublet:
(i) The two lenses must be of different materials.
Since, if `omega_1=omega_2`, `1/(f_1)+1/(f_2) = 0`,
i.e., `1/F=0` `=>` `F=oo`
i.e., combination will not behave as a lens, but as a plane glass plate.
(ii) As `omega_1` and `omega_2` are positive quantities, for eq. (6) to hold, `f_1` and `f_2` must be of opposite nature, i.e. if one of the lenses is convex the other must be concave.
(iii) If the achromatic combination is convergent
`f_C < f_D` and as - `(f_C)/(f_D)=(omega_C)/(omega_D)`, `omega_C < omega_D`
i.e. , a convergent achromatic doublet, convex lens has lesser focal length and dispersive power than divergent one.
For lenses separated by a distance: d = `((omega_2f_1)+(omega_1f_2))/(omega_1 + omega_2)`
In case of two thin lenses in contact
`1/F=1/(f_1)-1/(f_2)`
`=>` `-(dF)/(F^2)=-(df_1)/(f_1^2)-(df_2)/(f_2^2)`
The combination will be free from chromatic aberration if `dF = 0`
i.e., `(df_1)/(f_1^2) + (df_2)/(f_2^2)=0`
which in the light of equation (5) reduces to
`(omega_1f_1)/(f_1^2) + (omega_2f_2)/(f_2^2)`
i.e., `(omega_1)/(f_1) + (omega_2)/(f_2)=0............(6)`
This condition is called condition of achromatism (for two thin lenses in contact) and the lens combination which satisfies this condition achromatic lens. From this condition, i.e., from equation (6) it is clear that in case of achromatic doublet:
(i) The two lenses must be of different materials.
Since, if `omega_1=omega_2`, `1/(f_1)+1/(f_2) = 0`,
i.e., `1/F=0` `=>` `F=oo`
i.e., combination will not behave as a lens, but as a plane glass plate.
(ii) As `omega_1` and `omega_2` are positive quantities, for eq. (6) to hold, `f_1` and `f_2` must be of opposite nature, i.e. if one of the lenses is convex the other must be concave.
(iii) If the achromatic combination is convergent
`f_C < f_D` and as - `(f_C)/(f_D)=(omega_C)/(omega_D)`, `omega_C < omega_D`
i.e. , a convergent achromatic doublet, convex lens has lesser focal length and dispersive power than divergent one.
For lenses separated by a distance: d = `((omega_2f_1)+(omega_1f_2))/(omega_1 + omega_2)`