Lasers and Laser Light — Question 2

Question 2

Unequal beam powers reduce contrast even for perfectly coherent light

Two parallel-polarized beams interfere with intensities $I_1,I_2$ and coherence magnitude $g$. Their extrema are $I_{\max,\min}=I_1+I_2\pm2g\sqrt{I_1I_2}$. Visibility is $(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$.

1. Derive the visibility formula.

2. If $I_2=4I_1$ and measured visibility is .60, infer g and the visibility obtainable after balancing the powers without changing coherence.

Question 2 - Solution

Unequal beam powers reduce contrast even for perfectly coherent light

(a) Method and calculation

$$I_{\max}-I_{\min}=4g\sqrt{I_1I_2},\quad I_{\max}+I_{\min}=2(I_1+I_2).$$$$\boxed{\mathcal V=\frac{2g\sqrt{I_1I_2}}{I_1+I_2}}.$$

(b) Interpretation and further calculation

$$I_2=4I_1:\quad .60=\frac{4g}{5}\Rightarrow\boxed{g=.75}.$$$$I_1=I_2:\quad\boxed{\mathcal V_{\mathrm{balanced}}=g=.75}.$$

Physical check

Low contrast alone does not prove poor coherence. Power imbalance and polarization mismatch must be separated from temporal-coherence effects.