Polarization — Question 7

Question 7

A quarter-wave plate converts equal in-phase components into a circle

A lossless plate preserves x and y amplitudes but adds pi/2 to the phase of y relative to x. The incident beam is linearly polarized at 45 degrees, with total electric amplitude E0.

1. Write the incident and outgoing component fields and identify the outgoing polarization.

2. Find the transmitted power fraction and the fraction passing a subsequent linear analyzer at any angle.

Question 7 - Solution

A quarter-wave plate converts equal in-phase components into a circle

(a) Method and calculation

$$\mathbf E_{\mathrm{in}}=\frac{E_0}{\sqrt2}(\hat x+\hat y)\cos\omega t.$$$$\boxed{\mathbf E_{\mathrm{out}}=\frac{E_0}{\sqrt2}(\hat x\cos\omega t-\hat y\sin\omega t)},\quad |\mathbf E_{\mathrm{out}}|^2=E_0^2/2.$$

(b) Interpretation and further calculation

$$\boxed{I_{\mathrm{out}}=I_{\mathrm{in}}=\tfrac12\epsilon_0cE_0^2}.$$$$I_\theta=\tfrac14\epsilon_0cE_0^2(\cos^2\theta+\sin^2\theta)\Rightarrow\boxed{I_\theta/I_{\mathrm{out}}=1/2}.$$

Physical check

The field tip moves on a circle. A phase-only plate preserves average power even though instantaneous field magnitude changes.