Path Difference

Two waves, each emitted by the same source, can reach a point by travelling different paths. When this happens, Interference can occur.

There are two types of Path Difference :-

1. Geometrical Path Difference

2. Optical Path Difference

Geometrical Path Difference ( Higher Recap )

The Path Difference used within the Higher Physics course should actually be referred to as the Geometrical Path Difference. This is a path difference caused by a physical difference in the route taken by the two light beams. 

In the above diagram, the length of path 1 is clearly longer than the length of path 2, with a 

Path difference = Path 1 - Path 2 

Optical Path Difference 

When a beam of light passes from one medium into another of different refractive index, refraction can occur. This refraction causes the wave to change Velocity and therefore change Wavelength. This means that beams with the same physical distance travel can have a path difference as each will have travelled a different number of Wavelengths. Because of this, the Optical path difference is usually given in terms of wavelength. 

The above diagram shows two identical beams of light (ray colour for ease of description only) travelling left to right, in Phase. 

1. The Blue ray passes only through air 

2. The Red ray passes from the air into a glass block of length L then returns to the air

As can be seen from the diagram, the two rays are no longer in Phase after one passes through the glass. 

The Optical path difference caused the the refraction within the block is derived in the following way:-

Optical Path Length =  ( refractive index ) x ( Geometrical Path Length ) 

O.P.L ( Blue Ray )  =  nair x L 

O.P.L ( Red Ray )   =  nglass x L 

Optical Path Difference =  nglassL  - nairL

 =  nglassL - L                 (as nair = 1)

As seen in the Higher course, path difference is what causes Interference. 

For the above diagram :-

To be in phase :-

Path difference = ( S2Q - S1Q ) = mλ                     where m is an integer

To be out of phase :-

Path difference = ( S2Q - S1Q ) = ( m + 1/2 )λ       where m is an integer

Note - The above applies to both Geometrical and Optical Path Differences.

Phase Difference and Path Difference

The Phase difference is related to the path difference by :-

where :-

λ  - The wavelength in a vacuum ( m )