# Derivation of intensity at the central point in single slit diffraction

The legend himself, John Rennie, spent several hours in chat to convince me that if the slit width in a single slit diffraction pattern is doubled, then the intensity at the central point becomes $$4$$ times. Although I agree with , (and understand) his method, I have a different perspective:

The intensity is the magnitude of the Poynting vector: $$E^2/2\mu_{0}c$$ where $$E$$ is the electric field amplitude. The fact that intensity at the central point becomes $$4$$ times, suggests that the Amplitude of the resultant wave which forms at the central point, doubles upon doubling the slit.

Is there some mathematics to show this?

My textbook simply derives the intensity variation relative to the central point, and so it doesn't quite explain what happens to the electric field amplitude at the central point.

• I wrote about this a while ago in this post. In particular, I try to explain how you can reason this through in terms of amplitude and in terms of intensity and how these distinct lines of thought yield the same answer. Is the post helpful or is there something else you're looking for? – Maximal Ideal Aug 6 at 17:57
• thanks, your answer seems to be perfect. Just one question, is it really rigorous to say sinc(0)=1? – satan 29 Aug 6 at 18:12
• Yes, actually. The function $\operatorname{sinc} (x)$ is defined as $$\operatorname{sinc} (x) = \begin{cases} \frac{\sin x}{x} &\text{ if } x\ne 0, \\ 1 &\text{ if } x=0. \end{cases}$$ This is taken by definition, and then one can prove that the resulting function is continuous everywhere. – Maximal Ideal Aug 6 at 18:18