Deriving F=a2/λL in Fresnel Diffraction

[khanley]

Hi guys,

I can't find my notes on Fresnel diffraction and I have my Optics exam on Monday...

Does anyone know how to derive F=a²/λL where L=1/z+1/z'

I've googled it to death, checked here and tried in Optics 4th edition by Hecht with no luck...

I'd really appreciate any help.

Thank you so much!

 

[Niles]

I'll give it a try, but its only an approximate derivation. Maybe it will point you in the correct direction: So we are looking at a slit of width 2a, and a point source S placed some distance behind it. The distance from S to the upper/lower part of the slit we denote R.

The distance between the opening and R we denote Δ, and is the difference between the plane wavefront to the curved wavefront. We find

$$\Delta = R-\sqrt{R^2-a^2} = R-R\sqrt{1-\frac{a^2}{R^2}}\approx \frac{a^2}{2R}$$

If the wavefront is plane we must have Δ<<λ so R >> a²/λ. This is the Fraunhofer regime, so the Fresnel regime is R ~ a²/λ.

I hope it helps.