Question

Explain what is optical path length. How is it different from actual path length?

Answer 1

Hint: Optical path length can be given as the product of the refractive index and the distance which would have travelled by light at a particular time. The actual path length is the distance
travelled by light in a medium.

Formula used:
We will be using the following formulae; $$n={\displaystyle \frac{v_v}{v_m}}$$ where $$n$$ is the index of refraction of a particular medium, $$v_v$$ is the speed of light vacuum, $$v_m$$ is the speed of light in that medium. $$d_o=n\times d_g$$ where $$d_o$$ is the optical path length, $$n$$ is the refractive index of the medium and $$d_g$$ is the actual path length or the geometric length.

Complete step by step solution:
Generally optical path length and actual path length are related but are not identical.
The optical path length of light in a medium can actually be defined as the length or distance in which the light would have travelled in the same time if it were travelling in a vacuum. This statement means that if a light travels through a particular medium, it would have travelled a particular length or distance within a particular time, but the optical path length is the length or distance that light waves would have travelled if it were not travelling in a medium but in vacuum.
Generally, the optical length can be given as
$$d_o=n\times d_g$$ where $$d_o$$ is the optical path length, $$n$$ is the refractive index of the medium and $$d_g$$ is the actual path length or the geometric.
Actual path length or Geometric length is simply the real length travelled by the light in the medium.

Note: For clarity, we can prove the formula for the optical path length as follows:
The distance travelled by light in the medium is
$$d_g=v_m\times t$$
but now, the distance the length would have travelled in vacuum in the same time would be
$$d_o=v_v\times t$$
$$\Rightarrow t={\displaystyle \frac{d_o}{v_v}}$$
Inserting into $$d_g=v_m\times t$$
$$d_g=v_m\times {\displaystyle \frac{d_o}{v_v}}$$
By making $$d_o$$ subject of the formula,
$$d_o={\displaystyle \frac{v_v}{v_m}}{d}_g$$
Now recalling that, $$n={\displaystyle \frac{v_v}{v_m}}$$ where $$n$$ is the index of refraction of a particular medium, $$v_v$$ is the speed of light vacuum, $$v_m$$ is the speed of light in that medium, then
$$d_o=n{d}_g$$