Diffraction by a simple edge

The diffraction of light by a single straight edge is described by a general theory that must take into account the curvature of the wavefront. Unlike Fraunhofer diffraction at a single slit, Fresnel diffraction requires the calculation of the Fresnel integrals. A simple way to represent them is with the Cornu spiral. The cosine integral $\cal C$ is plotted on the horizontal axis, and the sine integral $\cal S$ is plotted on the vertical axis. The distance along the curve is called $w$. Notice that the spiral wraps around two points, at $(-1/2,-1/2)$ and $(1/2,1/2)$. The line drawn from the lower of these two asymptotes to the curve at $w$ is proportional to the amplitude of the diffracted light at a point $x$. The intensity is given by the square of the length of this line according to

$$I(w) = I_0/2 \left(\left({\cal C}\left(w\right) + \frac{1}{2}\right)^2 + \left({\cal S}\left(w\right) + \frac{1}{2}\right)^2\right)$$ (91)

Let $a$ be the distance from the light source to the edge, and let $b$ be the distance from the edge to the point of observation. Then $x$ is given simply by

$$x^2 = \left(b\lambda \left(a + b\right) / 2a\right) w^2$$ (92)

The intensity oscillations that mark a distinctive diffraction pattern occur for values of $w$ that cause the line segment to terminate where the spiral circles the upper asymptote at $(1/2,1/2)$.