# Varying the Relative Amounts of Light Passing Through Each Slit in the Double Slit Experiment

What happens if different amounts of light pass through each slit in the double slit experiment? For example, if twice as much light passed through one slit as through the other slit, how would the interference be affected. Could this phenomena be predicted with the Englert-Greenberger-Yasin duality relation?

• How do you know twice as much light passed though? – Viktor Nov 18 '15 at 4:13
• Well you could polarize the laser. Then you could attach two polarizer, the first which changes the polarization to 45 degrees with respect to the initial polarization and a second which would bring the polarization back to the initial polarization, to each other and put then over one of the slits. In general, equal amounts of light pass through each slit, so the transmitted light from the slit with the polarizers would have half the intensity of the second slit according to Malus' law. – Clement Decker Nov 18 '15 at 4:44

The two patterns are some complex-valued functions $\psi_{1,2}(z)$ on the screen which you only see in absolute value, $|\psi_{1,2}(z)|^2$ being the actual pattern seen. Typically for example you might have $$\psi_\pm(z) = {(2\pi\sigma^2)}^{1/4}\exp\left[\frac{(z \pm \mu)^2}{4\sigma^2}\right] e^{i k (x \pm \mu)}$$for the two functions; you can confirm that $|\psi_\pm(z)|^2$ are both Gaussian probability distributions with unit integrals, so the light appears in two bell curves centered each at $z = \mp \mu$ with characteristic width $\sigma.$
As for the superposition, let $I_+$ be the intensity transmitted through $\psi_+$ and $I_-$ be the intensity transmitted through $\psi_-,$ then the overlap in the double-slit experiment is simply $$I(z) = {\left|\sqrt{I_+}~\psi_+(z) + \sqrt{I_-}~\psi_-(z)\right|}^2.$$