Question

Answer in brief:

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

What is Optical Path length? How is it different from the actual Path length? 

Answer 1

Consider, a light wave with an angular frequency of w and a wave vector of k travelling in the x-direction through a vacuum. The phase of this wave is (kx - ωt). In a vacuum, light speed at c, but in a medium, it speeds at v.

k = `(2pi)/lambda = (2pi v)/(v lambda) = omega/v` as ω = 2πv and v = vλ, where v is the frequency of light.

If the wave travels a distance Δ x, its phase changes by Δ Φ = kΔx = ω Δx/v.

Similarly, if the wave is travelling in vacuum,

k = ω/c and Δ Φ = ω Δ x/c

Now, consider a wave travelling a distance Δ x in the medium, the phase difference generated is,

Δ Φ' = k' Δ x = ωn Δ x/c = ω Δ x'/c     ...(1)

where Δ x' = n Δ x              .....(2)

The distance nΔ x is called the optical path length of the light in the medium; it is the distance the light would have travelled in the same time t in vacuum (with the speed c).

The optical path length in a medium is the corresponding path in a vacuum that light traverses at the same time as it does in the medium.

Now, speed = `"distance"/"time"`

∴ time = `"distance"/"speed"`

∴ t = `"d"_"medium"/"v"_"medium" = "d"_"vaccum"/"v"_"vaccum"`

Hence, the optical path = `"d"_"vacuum"`

`= "v"_"vaccum"/"v"_"medium" xx "d"_"medium"`

`= "n" xx "d"_"medium"`

Thus, a distance d travelled in a medium of refractive index n introduces a path difference = nd - d = d (n - 1) over a ray travelling equal distance through vacuum.

Answer 2

i. When a wave travels a distance Δx through a medium having a refractive index of n, its phase changes by the same amount as it would if the wave had travelled a distance nΔx in a vacuum. 

ii. Thus, a path length of Δx in a medium of refractive index n is equivalent to a path length of nΔx in a vacuum.

iii. nΔx is called the optical path travelled by a wave.

iv. This means, the optical path through a medium is the effective path travelled by light in a vacuum to generate the same phase difference.

v. Optical path in a medium can also be defined as the corresponding path in a vacuum that the light travels at the same time as it takes in the given medium.

i.e., time = `"d"_"medium"/"v"_"medium" = "d"_"vacuum"/"v"_"vacuum"`

∴ `"d"_"vacuum" = "v"_"vacuum"/"v"_"medium" xx "d"_"medium" = "n" xx "d"_"medium"`

But `"d"_"vacuum"` = Optical path  

∴ Optical path = n × `"d"_"medium"`

Thus, a distance d travelled in a medium of refractive index n introduces a path difference = nd - d = d(n - 1) over a ray travelling an equal distance through the vacuum.