# What is the theory behind Fresnel's half period zones?

I was not convinced by exactly how does this work,why do we need to take concentric spheres around point P and how do those spheres cast circular areas on the wavefront? Why is the radius OM1 and the first sphere originating from P not same?

• Which book is this Jan 8 '20 at 11:43

Consider the wavefronts (orange) of a plane wave moving in from the left.

Each point on a wavefront acts as a secondary source - Huygen's construction.

To get to point $$F$$ the wavelets from point $$X$$ on the incident wavefront have to travel a distance $$XF$$.

The wavelets from a point $$Y$$ on the incident wavefront travel a distance $$YF$$ and so do not arrive in phase with the wavelets from point $$X$$.
At point $$F$$ there is a superposition of these two sets of wavelets.
As long as the path difference between the two sets of wavelets is less than $$\frac \lambda 2$$, where $$\lambda$$ is the wavelength of the incident waves the wavelets will superpose constructively ie combine together to give a greater amplitude of the wave at $$F$$.

If $$AF$$ and $$A'F$$ differ in distance by less than $$\lambda 2$$ then the wavelets which originate from the wavefront $$AA'$$ will arrive with a phase difference of less than $$\frac \lambda 2$$ and add together constructively.

Now consider the incident wavefront between $$AB$$ and consider a point $$Z$$ producing wavelets.
The distance $$ZF$$ is larger that $$XF$$ by at least $$\frac \lambda 2$$ and so the wavelet originate from $$Z$$ will be out of phase by greater than $$\frac \lambda 2$$ and so when they arrive at $$F$$ will superpose in a destructive manner reducing the amplitude of the resultant wave at $$F$$.
So that this not happen all waves which would produce a phase difference between $$\frac \lambda 2\,(A)$$ and $$\lambda \, (B)$$, and between $$\frac \lambda 2\, (A')$$ and $$\lambda \,(B')$$ are removed by making $$AB$$ and $$A'B'$$ opaque.

The next zone defined by $$BC$$ and $$B'C'$$ is made transparent because all the waves which arrive from that zone are between $$\lambda$$ and $$\frac{3\lambda}{2}$$ out of phase with the waves which came directly from $$X$$ and so will superpose constructively.

The end result is the constructive superposition of wave at $$F$$.

My diagram is just a section through the zone plate which has rotational symmetry about the line $$XF$$.

The radii of the circles which define the transparent and opaque zones do not increase in equal increments, just think about the geometry of the situation.

And finally a zone plate can be constructed with the transparent and opaque sections transposed.

It's just commonly known diffraction in 2d. (https://en.wikipedia.org/wiki/Diffraction) The expression $$d\sin(\theta_n)=n\lambda$$ in polar symmetry.