Diffraction — Question 3

Question 3

Fresnel number distinguishes two observation distances

Define the Fresnel number using the full slit width by $F=a^2/(\lambda L)$. A slit has $a=0.40\,\mathrm{mm}$ and is illuminated at $\lambda=640\,\mathrm{nm}$. Compare screens at $L_A=0.10\,\mathrm m$ and $L_B=10\,\mathrm m$. For this problem, call $F\leq0.1$ a conservative far-field condition.

1. Calculate both Fresnel numbers and classify the screens under the stated criterion.

2. Find the smallest screen distance satisfying that criterion and explain the effect of doubling a.

Question 3 - Solution

Fresnel number distinguishes two observation distances

(a) Method and calculation

$$\boxed{F_A=\frac{(4.0\times10^{-4})^2}{(6.4\times10^{-7})(0.10)}=2.50}.$$$$\boxed{F_B=0.0250};\quad F_A>0.1,\quad F_B\leq0.1.$$

(b) Interpretation and further calculation

$$F\leq0.1\Longrightarrow\boxed{L\geq\frac{10a^2}{\lambda}=2.50\,\mathrm m}.$$$$a\to2a\Longrightarrow L_{\min}\to4L_{\min}=\boxed{10.0\,\mathrm m}.$$

Physical check

The transition is gradual; the numerical threshold is an explicit accuracy convention rather than a sharp physical boundary.