Intensity in Double-Slit Interference — Question 1

Question 1

Adding field phasors gives the intensity at an arbitrary phase

Two coherent, parallel-polarized waves in the same medium have peak electric-field amplitudes $E_0$ and $2E_0$. Define $I_0$ as the intensity of the first wave alone. Their relative phase is $\delta=120^\circ$.

1. Add the field phasors and obtain the intensity at this phase.

2. Find the maximum and minimum possible intensities and their visibility as the phase is varied.

Question 1 - Solution

Adding field phasors gives the intensity at an arbitrary phase

(a) Method and calculation

$$\widetilde E=E_0(1+2e^{i\delta}),\quad I/I_0=|1+2e^{i\delta}|^2=5+4\cos\delta.$$$$\boxed{I(120^\circ)=3I_0}.$$

(b) Interpretation and further calculation

$$\boxed{I_{\max}=(1+2)^2I_0=9I_0,\quad I_{\min}=(2-1)^2I_0=I_0}.$$$$\boxed{V=\frac{9-1}{9+1}=0.80}.$$

Physical check

Field amplitudes add before squaring. A phase difference of pi cannot fully cancel beams with unequal amplitudes.