Question

Two coherent light sources having intensity in the ratio $2x$ produce an interference pattern. The ratio $(I_{max} - I_{min}) / (I_{max} + I_{min})$ will be :

• A. $\frac{\sqrt{2x}}{2x+1}$
• B. $\frac{2\sqrt{2x}}{2x+1}$
• C. $\frac{\sqrt{2x}}{x+1}$
• D. $\frac{2\sqrt{2x}}{x+1}$

Hint:

The expression $\frac{I_{max} - I_{min}}{I_{max} + I_{min}}$ is also known as the {Fringe Visibility}. It reaches a maximum value of 1 when the two sources have identical intensities ($I_1 = I_2$).

Answer 1

The Correct Option is B

Step 1: Let $I_1/I_2 = 2x$. The ratio of amplitudes is $r = \sqrt{I_1/I_2} = \sqrt{2x}$.
Step 2: Visibility/Interference ratio is given by $\frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{2\sqrt{I_1 I_2}}{I_1 + I_2}$.
Step 3: Divide numerator and denominator by $I_2$: $\frac{2\sqrt{I_1/I_2}}{I_1/I_2 + 1}$.
Step 4: Substitute $I_1/I_2 = 2x$: $\frac{2\sqrt{2x}}{2x + 1}$.