The defect in image produced in the formation of image of an axial point object (of monochromatic light) by a spherical mirror or lens is called spherical aberration. The image of an object in point object formed by a spherical mirror or by a spherical lens is usually blurred. This defect of image is called spherical aberration
Methods to reduce spherical aberration :

(a) `text(For Mirrors :)`
By using a proper surface e.g., paraboloidal surface for parallel incident beam.

(b) `text(For Lenses :)`
In lenses spherical aberration cannot be completely vanished. It can be minimized only.
(i) By using stops.
(ii) By using crossed lens.
Note : for minimum spherical aberration.

`(R_1)/(R_2) = (2mu^2 -mu-4)/(mu(2mu-1))`

(iii) By using combination of lenses, `d = f_1 - f_2`.

The image of an object in white light formed by a lens is usually coloured and blurred. This defect of image is called chromatic aberration and arises due to the fact that focal length of a lens is different for different colours. Fora single lens,

`1/f = (mu-1)[1/(R_1) - 1/(R_2)]`

and an `mu` of lens in maximum for violet while minimum for red, violet is focused nearest to the lens while red farthest from it as shown in figure.
As a result of this in case of convergent lens, at `F_V` centre of image will be violet and focused while sides red and blurred while at `F_R` reverse is the case, i.e., centre will be red and focused while sides violet and blurred. The difference between `f_V` and `f_R` is a measure of longitudinal chromatic aberration, i.e.,

L.C.A. = `f_R - f_V=-df`
`=>` `df=f_V-f_R..........(1)`

However, as for a single lens,

`1/f = (mu-1)[1/(R_1) - 1/(R_2)]...............(2)`

i.e., `-(df)/(f^2)=dmu[1/(R_1) - 1/(R_2)]...........(3)`

So dividing equation (3) by (2)

`-(df)/f = (dmu)/(mu-1)=omega............(4)` [as `omega= (dmu)/(mu-1)`]

And hence, from equation (1) and (4),

L.C.A. `=- df = omegaf........(5)`

Now, as for a single lens neither f nor `omega` can be zero, we cannot zero, we cannot have a single lens free from chromatic aberration.

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)`