Chromatic aberration ..in lenses

It is known  that  focal length of a lens depends on the refractive index of the material of the lens. From the lens maker's  formula  it can be easily  concluded  that  for  greater refractive index  lenses have lesser focal length in the magnitude . It is also well known that violet color has maximum ref index and red least.  So a lens will have least focal length  ( in magnitude ) for violet and maximum for red. For others  follow the order in VIBGYOR.

So if an object is emitting white light the images of different color will be formed at different places. This is not desired and this defect is called chromatic aberration.
The numerical value of the chromatic aberration is defined as the separation  between the images of violet and red color .  Dispersive power  ω of a lens is defined as  the ratio of the difference between  the focal lengths of red and violet color to the mean focal length which is for yellow.

ω =( fr - fv )/ fy 

It can be proved that when lenses  ( say A  and B ) of opposite nature i.e. on convex and other concave are placed in contact , then the combination will act as non chromatic i.e.achromatic, if their mean focal lengths and the dispersive powers follow the  relation     ω_A/  ω_{B  =} - f_A / f_B.

_{The non chromatic ( achromatic)  combination of lenses of same nature can also be obtained if they are placed with separation  d  which satisfy the relation  :  d = (} ω_A f_B +  ω_B f_{A ) / (} ω_A+ω_{B )}