Refraction Of Light

Refraction Of Light

OPTICAL PATH (\( \Delta \)x):
The shortest distance between any two points A and B is called geometrical path. The length of geometric path is independent of the medium that surrounds the path AB. When a light ray travels from the point A to point B it travels with the velocity c if  the medium is vacuum and with a lesser velocity v if the medium is other than vacuum. Therefore the light ray takes more time to go from A to B located in a medium.
    The optical path to a given geometrical path in a given medium is defined as distance travelled by  light in vacuum in the same time in which it a given path length in that medium.

AB = real path or geometrical path
A'B' = optical path
If the light travels a path length 'd' in a medium at speed v, the time taken by it will be  \( \left( {\frac{d} {v}} \right) \)
So optical path length,
    \( \Delta \)x=c\( \times \)t=C\( \times \)\( \left( {\frac{d} {v}} \right) \)=(as \( \mu \)=\( \frac{c} {v} \)
    Therefore optical path is µ times the geome trical path. As for all media µ>1, optical path length is always greater than actual path length.

Note:
If in a given time t, light has same optical path length in different media, and if light travels a distance \( d_1 \) in a medium of refractive index \( \mu _1 \), and a distance \( d_2 \) in a medium of refractive index \( \mu _2 \) in same time t, then
        \( \mu _1 \)\( d_1 \)    =\( \mu _2 \)\( d_2 \)
Note:
The difference in distance travelled by light in vacuum and in a medium in the same interval of time is called optical path difference due to that medium. 
\( \Delta \)r=A'B'-AB = µd-d   \( \Delta \)x=(µ-1)