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Interference of Light

Problem 5.94

In Michelson's interferometer the yellow sodium line composed of two wavelengths λ1=589.0nm\lambda_{1}=589.0 \mathrm{nm} and λ2=589.6nm\lambda_{2}=589.6 \mathrm{nm} was used. In the process of translational displacement of one of the mirrors the interference pattern vanished periodically (why?). Find the displacement of the mirror between two successive appearances of the sharpest pattern.

Reveal Answer
5.94. The transition from one sharp pattern to another occurs if the following condition is met: (k+1)λ1=kλ2 (k+1) \lambda_{1}=k \lambda_{2} where kk is a certain integer. The corresponding displacement Δh\Delta h of the mirror is determined from the equation 2Δh=kλ^2.2 \Delta h=k \hat{\lambda}_{2} . From these two equations we get Δh=λ1λ22(λ2−λ1)≈λ22Δλ=0.3 mm \Delta h=\frac{\lambda_{1} \lambda_{2}}{2\left(\lambda_{2}-\lambda_{1}\right)} \approx \frac{\lambda^{2}}{2 \Delta \lambda}=0.3 \mathrm{~mm}
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