Question

Figure 35-29 shows the transmission of light through a thin film in the air by a perpendicular beam (tilted in the figure for clarity). (a) Did ray${\mathit{r}}_{\mathbf{3}}$undergo a phase shift due to reflection? (b) In wavelengths, what is the reflection phase shift for ray${\mathit{r}}_{\mathbf{4}}$? (c) If the film thickness is L, what is the path length difference between rays${\mathit{r}}_{\mathbf{3}}$and${\mathit{r}}_{\mathbf{4}}$?

Short answer

(a) There is no phase shift due to reflection in the ray$r_3$.

(b) The reflection phase shift for a ray $r_4$is 0.

(c) The path length difference between rays $r_3$and$r_4$is 2L.

Step-by-step solution

Given information 

The thickness of the thin film is, L

The two light rays passing through the thin film are, $r_3$and$r_4$.

Reflection shift 

When a light ray reflects from a medium with a higher value of refraction index, then a phase change of 180 degrees happens, and when it reflects from a medium with a smaller refraction index, no phase change happens.

If the light ray passes through a medium of thickness L without any reflection then its path length would also be equal to L.

Step 3(a): Reflection phase shift for ray 3 

From the diagram, it is clear that one incident ray gets refracted and passes through the thin film coming out of the other side of the film.

The path of the ray$r_3$is the same as the incident ray, it means that there is no phase shift that happens due to the reflection in the ray$r_3$.

Hence, there is no phase shift due to reflection in the rayrole="math" localid="1663144835974" $r_3$.

Step 4(b): Reflection phase shift for ray 4 

The incident ray gets reflected two times inside the thin film and then passes out from the other side of the film.

If the direction of the refracted and reflected ray$r_4$does not change, then its value of reflection phase shift would also be zero.

Hence, the reflection phase shift for a ray$r_4$is 0.

Step 5(c): Path length difference between rays 3 and 4 

According to the question, the light beam istransmitted through a thin film in a perpendicular direction.

So, the formula for the path length difference between rays $r_3$and $r_4$ is given by,

Path length difference = Path length of $r_4$- path length of$r_3$

Path length difference = (L+L+L)-L

Path length difference = 3L-L

Path length difference = 2L

Hence, the path length difference between rays$r_3$androle="math" localid="1663144064356" $r_4$is 2L.