Question

Show that all Fresnel half-period zones have the same area.

Answer 1

Answer on Question #39082, Physics, Optics

Show that all Fresnel half-period zones have the same area.

Solution.

Let the light source $S$ extends monochromatic spherical wave, $P$ - the point of observation. Through the point $O$ passes spherical wave surface. It is symmetric with respect to the line $SP$ .

To determine the resultant effect at $P$ , Fresnel subdivided the wavefront into a number of circular zones I, II, III etc. Let $PO = b$ . The radii of zones equal $b + \frac{\lambda}{2}, b + \frac{2\lambda}{2}, b + \frac{3\lambda}{2}$ ... etc. The area enclosed between O and $O_1$ , $O_1$ and $O_2$ , $O_2$ and $O_3$ etc. are known as half period zones. Each zone differs from its neighbour by a phase difference of $\pi$ or a path difference of $\lambda / 2$ . The area enclosed by the first circle of radius $OO_1$ is called the first half period zone. The area enclosed by the annular strip $O_1O_2$ is known as second half period zone and so on. Thus, the annular area between (n-1)th circle and nth is the nth half period zone.

The area of the nth zone:

$$\begin{array}{l} S _ {n} = \pi O O _ {n} ^ {2} - O O _ {n - 1} ^ {2} = \pi \left[ \left(P O _ {n} ^ {2} - P O ^ {2}\right) - \left(P O _ {n - 1} ^ {2} - P O ^ {2}\right) \right] = \pi \left[ P O _ {n} ^ {2} - P O _ {n - 1} ^ {2} \right] = \\ = \pi \left[ \left(b + \frac {n \lambda}{2}\right) ^ {2} - \left(b + \frac {(n - 1) \lambda}{2}\right) ^ {2} \right] = \\ = \pi \left[ b ^ {2} + \frac {n ^ {2} \lambda^ {2}}{4} + b n \lambda - b ^ {2} - \frac {(n - 1) ^ {2} \lambda^ {2}}{4} - b \lambda (n - 1) \right] = \\ = \pi \left[ b \lambda + \frac {\lambda^ {2}}{4} (n ^ {2} - n ^ {2} - 1 + 2 n) \right] = \pi \left[ b \lambda + \frac {\lambda^ {2}}{4} (2 n - 1) \right] \\ \end{array}$$

$b \gg \lambda$ so $\lambda^2$ term is negligible.

Thus,

$$S _ {n} \approx \pi b \lambda .$$

The area of each half period zone $(\pi b\lambda)$ is approximately same and independent of the order of zone.