Diffraction — Question 6

Question 6

A straight-edge shadow acquires a wavelength-dependent transition scale

For an introductory straight-edge model, define the transverse diffraction scale by $w=\sqrt{\lambda L}$, where L is distance beyond the edge. This is a scale estimate, not a definition of a particular intensity contour. Use $\lambda=500\,\mathrm{nm}$ and $L=0.80\,\mathrm m$.

1. Calculate w and the distance needed to make w equal to $2.0\,\mathrm{mm}$.

2. Find how L must change if wavelength is doubled while the same w is required.

Question 6 - Solution

A straight-edge shadow acquires a wavelength-dependent transition scale

(a) Method and calculation

$$\boxed{w=\sqrt{(5.0\times10^{-7})(0.80)}=0.63246\,\mathrm{mm}}.$$$$L=w^2/\lambda\Longrightarrow\boxed{L=8.0\,\mathrm m\ \text{for}\ w=2.0\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$w^2=\lambda L=\lambda^\prime L^\prime,\qquad\lambda^\prime=2\lambda.$$$$\boxed{L^\prime=L/2}.$$

Physical check

The square-root growth of the edge scale differs from the linear screen-distance growth of a fixed far-field angular pattern.