A parallel beam of light from a laser falls on a solid transparent sphere of index of refraction n, as shown in the figure. (a) We have to show that the beam cannot be brought to a focus at the back of the sphere unless the beam width is small compared with the radius of the sphere. (b) If the condition in (a) is satisfied, we have to find the index of refraction of the sphere. (c) We have to find the index of refraction, if any, that will focus the beam at the centre of the sphere.
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Solution: Click For PDF Version (a)
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As the index of refraction of the material of the spherical block is n, we have from Snell’s law
Also, we note that
If
and the following condition for focussing of the beam can be satisfied:
(b) This condition fixes the value of the index of refraction n for which the beam with half-width AB will get focussed at P. It will be given by the relation
(c) The condition for the beam to be focussed at the centre of curvature of the sphere is that the angle of refraction should be equal to zero. That is
As
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e
consider a beam with half-width AB,
as shown in the figure. The ray that strikes the spherical surface
at A is considered so that it is focused at P
, the point at the back of the sphere. From the line diagram, we
note that the incident angle
,
angle of refraction
and the angle
that the refracted ray makes with the axis at the point P
are related as
.
.
.
,
the approximation will be better and the incident angles and angle
of refraction will be small. Beam with small half-width compared to
the radius of curvature R
can be focussed at P
, as then
,
,
this condition cannot be satisfied for any finite value of n.