Lasers and Laser Light — Question 9

Question 9

Polarization mismatch removes interference without broadening either laser spectrum

Two mutually coherent monochromatic beams have equal intensity $I_0$ and linear polarization unit vectors $\hat e_1,\hat e_2$ separated by angle $\alpha$. Their relative optical phase is $\phi$.

1. Derive their intensity and fringe visibility.

2. For orthogonal input polarizations, insert an ideal polarizer at 45 degrees to both and find the transmitted mean intensity and visibility.

Question 9 - Solution

Polarization mismatch removes interference without broadening either laser spectrum

(a) Method and calculation

$$I=I_0+I_0+2I_0(\hat e_1\cdot\hat e_2)\cos\phi=\boxed{2I_0[1+\cos\alpha\cos\phi]}.$$$$\boxed{\mathcal V=|\cos\alpha|};\quad\alpha=90^\circ\Rightarrow\mathcal V=0.$$

(b) Interpretation and further calculation

$$I_1\prime=I_2\prime=I_0\cos^245^\circ=I_0/2.$$$$\boxed{\bar I\prime=I_0,\quad I\prime=I_0(1+\cos\phi),\quad\mathcal V\prime=1}.$$

Physical check

The polarizer projects both fields onto the same polarization direction, restoring interference at the cost of average transmitted power. The initially absent fringes do not imply lost temporal coherence.