91Ó°ÊÓ

The wavelength of both light rays is,$\mathrm{λ}=620\text{nm}$

When two waves of equal frequency and opposite phase meet each other then they cancel each other out and results in the ‘destructive interference’.

For destructive interference, the value of the path difference between two waves is given by,

$\mathrm{Δ}x=\left(2n+1\right)\left(\frac{\mathrm{λ}}{2}\right)$

Here,$\mathrm{λ}$ is the wavelength of both waves and $n=1,2,3,...$is the number of fringes.

According to the question, the wave$W_2$reflects twice from that same mirror and once from a tiny mirror located at a distance $L$. Has no substantive effect on the calculations,

The value of the phase difference after two reflections will be,

$2\left(\frac{\mathrm{λ}}{2}\right)=\mathrm{λ}$

The distance travelled by wave $W_2$is $2L$greater than the distance travelled by wave $W_1$.

Then the smallest value of $L$that puts the wave role="math" localid="1663049389226" $W_2$a half-wavelength "behind" wave $W_1$is given by,

$$\begin{gathered} 2L=\frac{\mathrm{λ}}{2} \\ L=\frac{\mathrm{λ}}{4} \\ L=\frac{620\text{nm}}{4} \\ L=155\text{nm} \end{gathered}$$

Hence, the smallest value of$L$ that puts the final light waves exactly out of phase is$155\text{nm}$ .

The fact that wave $W_2$reflects two additional times has no substantive effect on the calculations. Since two reflections amount to a $2\left(\frac{\mathrm{λ}}{2}\right)=\mathrm{λ}$phase difference, which is effectively not a phase difference at all. The substantive difference between $W_2$and $W_1$is the extra distance 2L travelled by $W_2$. Destructive interference will again appear if$W_2$is$\frac{3\mathrm{λ}}{2}$"behind" the other wave.

In this case, $2L\text{'}=\frac{3\mathrm{λ}}{2}$and the difference is,

$$\begin{gathered} L\text{'}-L=\frac{3\mathrm{λ}}{4}-\frac{\mathrm{λ}}{4} \\ L\text{'}-L=\frac{\mathrm{λ}}{2} \\ L\text{'}-L=\frac{620\text{nm}}{2} \\ L\text{'}-L=310\text{nm} \end{gathered}$$

Hence, the tiny mirror should be moved $310\text{nm}$away from the bigger mirror to again put the final waves out of phase.