Chromatic, Spherical Aberration of Lenses and Achromatic Combination | Physics Grade XI

Aberration of Lenses

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Chromatic, Spherical Aberration of Lenses and Achromatic Combination | Physics Grade XI

Chromatic, Spherical Aberration of Lenses and Achromatic Combination

Chromatic Aberration in lenses
The inability of a lens to focus all the colors of light at a single point is called chromatic aberration or axial or longitudinal chromatic aberration. Th is measured by the difference in focal lengths between red and violet colors.

Chromatic aberration = fr - fv

Chromatic aberration in convex lensChromatic aberration in concave lensUsing lens maker formula, for mean color of light we have,
1/f = (µ-1) (1/R1 + 1/R2)
1/R1 + 1/R2 = 1/ f(µ-1) …….... (i)
Where f is focal length of mean color, µ is refractive index of mean color. R1 and R2 are radii of curvature of two lens surfaces.
 
For violet color we have,
1/fv = (µv -1) (1/R1 + 1/R2)
1/ fv = (µv -1). 1/ f (µ-1) …… (ii)
 
Similarly, for red color,
1/ fr = (µr -1). 1/ f (µ-1) ……. (iii)
Here, in equation (ii) and (iii), µv and µr are the refractive index for violet and red color respectively and fv and fr are the focal length of the violet and red color, respectively.
 
Subtracting equation (iii) from (ii),
1/fv – 1/fr = ((µv -1). 1 - (µr -1). 1) / f (µ-1)
(fr – fv) / (fv. fr) = (µv – 1 - µr + 1) / f (µ-1)
fr – fv = ((µv – 1 - µr + 1). fv. fr) / f (µ-1) ...........(iv)
 
Since focal lengths of seven colors are in geometric progression, so we can write
f = √ fv. fr
f2 = fv. fr
 
The equation (iv) becomes,
fr – fv = (µv – 1 - µr + 1). f2 / f (µ-1)
fr – fv = f (µv - µr)/ (µ-1)
fr – fv = ω. f
Hence, chromatic aberration is the product of dispersive power and focal length for mean color of light.
 
Achromatic Combination of Lenses
The combination of two thin lenses in which their combination is free from chromatic aberration is called the achromatic combination of lenses.

Achromatic combination of lenses

Consider two thin lenses L and L' of dispersive power ω and ω' respectively placed in contact with each other as shown in the figure. Let µv, µ and µr are the refractive indices of L for violet, mean and red color respectively, and fv, f and fr are the focal length of respective colors. Similarly, µv', µ', µr', fv', f' and fr' are corresponding quantities of L'.

For lens L, the focal length of mean color is
1/f = (µ -1) (1/r1 + 1/r2)
1/r1 + 1/r2 = 1 / f (µ -1)
Where r1 and r2 are the radii of the curvature of two surfaces,
 
For Focal length of lens L for violet color is
1/fv = (µv -1) (1/r1 + 1/r2)
1/fv = (µv -1) / f (µ-1) ….. (i)
 
Similarly, for focal length of lens ‘L'’ for violet color
1/ fv' = (µv' -1) / f' (µ'-1) ….. (ii)
 
If Fv is the combined focal length of two lenses for violet color, then
1/Fv = 1/fv + 1/fv' …… (iii)
or, 1/Fv = (µv -1)/f(µ-1) + (µv' -1)/f' (µ'-1) …...... (iv)
 
In the same way for red color,
1/Fr = (µr -1)/f(µ-1) + (µr' -1)/ f'(µ'-1) …....... (v)
 
For achromatic combination, we have
Fr = Fv
or, 1/Fv = 1/Fr
or, (µv -1)/f(µ-1) + (µv' -1)/f'(µ'-1) = (µr -1)/f(µ-1) + (µr' -1)/f' (µ'-1)
or, (µv -1)/f(µ-1) - (µr -1)/f(µ-1) = (µr' -1)/f'(µ'-1) - (µv' -1)/f' (µ'-1)
or, µv - µr/f(µ-1) = - (µv' - µr')/f'(µ'-1)
or, ω/f = - ω'/ f' [ since, (µv - µr)/(µ-1) = ω and (µv' - µr')/(µ'-1) = ω']
or, ω/f + ω'/ f' = 0
.: This is the condition for achromatic combination of two lenses.
 
Spherical Aberration in a Lens
If a point object O is placed on the axis of a large lens, images Im and Ip will be formed by the paraxial and marginal rays respectively. It is clear from the figure that the paraxial ray form image at longer distance than the marginal rays. The image is not sharp at any point. This fault of lens is called spherical aberration.

Spherical aberration in convex lensSpherical aberration in concave lens

Removal of Spherical Aberration in Lenses

  1. By using stop: Spherical aberration can be removed by either cutting off the paraxial rays or marginal rays.
  2. By using plano-convex lenses
  3. By using two lenses separated by a distance
  4. By combining suitable concave and convex lens
Scattering of light
The light is accelerated by the electric field due to which the electric charge emits radiation in all directions, and this process is called scattering.
 
The blue color of sky
When light from the sun reaches to earth’s atmosphere, the different wavelength of light gets scattered from their path following Rayleigh’s law that says
      Amount of scattering α 1/ λ4
Since the wavelength of blue light is almost half of the red light, the scattering of red light is 16 times more than the red light due to which the blue light predominates and the sky appears blue.

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