Polarization — Question 6

Question 6

A diagonal analyzer detects coherence missed by horizontal and vertical tests

A monochromatic vacuum beam has $E_x=a\cos\omega t$ and $E_y=b\cos(\omega t+\delta)$, with positive amplitudes a and b. An ideal analyzer transmits the projection along angle theta from x.

1. Derive the transmitted cycle-averaged intensity.

2. For a=b, find the 45-degree analyzer intensity for delta=0, pi/2, and pi in units of the total incident intensity.

Question 6 - Solution

A diagonal analyzer detects coherence missed by horizontal and vertical tests

(a) Method and calculation

$$E_\theta=a\cos\theta\cos\omega t+b\sin\theta\cos(\omega t+\delta).$$$$\boxed{I_\theta=\tfrac12\epsilon_0c[a^2\cos^2\theta+b^2\sin^2\theta+2ab\cos\delta\sin\theta\cos\theta]}.$$

(b) Interpretation and further calculation

$$a=b:\ I_{\mathrm{in}}=\epsilon_0ca^2,\quad I_{45}/I_{\mathrm{in}}=(1+\cos\delta)/2.$$$$\boxed{I_{45}/I_{\mathrm{in}}=1,\ 1/2,\ 0\quad\text{for}\quad\delta=0,\pi/2,\pi}.$$

Physical check

Horizontal and vertical analyzer readings are independent of delta; the diagonal cross term reveals relative-phase information.