12. Fraunhofer Approximation#
12.1. Example 1: A Square Aperture#
A \(100 \mu m \times 1 \mathrm{mm}\) aperture illuminated by a laser \(\lambda = 500 nm\).
Find the diffraction pattern.
Find the necessary distance away from the aperture.
Assume that the incident field is a plane wave
\[
E(x, y) = \frac{e^{j k z}}{j \lambda z} e^{j \frac{k}{2z}(x^2 + y^2)} \mathfrak{F}\{ P(x, y) \} |_{f_x = x / \lambda z, f_y = y / \lambda z}
\]
\[\begin{split}
P(x) = \left\{ \begin{matrix} 1 & \lvert x \rvert \leq 50 \mu m \\ 0 & \mathrm{otherwise} \end{matrix} \right.
\end{split}\]
This is similar to
\[\begin{split}
\mathrm{rect}(x) = \left\{ \begin{matrix} 1 & \lvert x \rvert \leq \frac{1}{2} \\ 0 & \mathrm{otherwise} \end{matrix} \right.
\end{split}\]
\[\begin{split}
\begin{align}
\begin{split}
\mathrm{rect}(a x) &= \left\{ \begin{matrix} 1 & \lvert a x \rvert \leq \frac{1}{2} \\ 0 & \mathrm{otherwise} \end{matrix} \right. \\
&= \left\{ \begin{matrix} 1 & \lvert x \rvert \leq \frac{1}{2a} \\ 0 & \mathrm{otherwise} \end{matrix} \right. \\
\end{split}
\end{align}
\end{split}\]
Let \( a = \frac{1}{100} \mu m \).
\[\begin{split}
P(x) = \mathrm{rect}(\frac{x}{100\mu m}) = \left\{ \begin{matrix} 1 & \lvert x \rvert \leq 50 \mu m \\ 0 & \mathrm{otherwise} \end{matrix} \right.
\end{split}\]
Now take the Fourier transform,
\[
\mathfrak{F}\{ \mathrm{rect}(ax) \} = \frac{1}{a} \mathrm{sinc} \left( \frac{F_x}{a} \right)
\]
\[
P(F_x) = 100 \times 10^{-6} \mathrm{sinc} (100 \times 10^{-6} F_x)
\]
Similar for the y-direction
\[
P(F_y) = 10^{-3} \mathrm{sinc} (10^{-3} F_y)
\]
where
\[
F_x = \frac{x}{\lambda z}, F_y = \frac{y}{\lambda z}
\]
Hence,
\[
E(x, y) = \frac{e^{j k z}}{j \lambda z} e^{j \frac{k}{2z} (x^2 + y^2)} (100 \times 10^{-6}) (10^{-3}) \mathrm{sinc} \left( \frac{100 \times 10^{-6} x}{\lambda z} \right) \mathrm{sinc} \left( \frac{10^{-3} y}{\lambda z} \right)
\]
Zeros are at
\[
\frac{100 \times 10^{-6} x}{\lambda z} = \pm 1
\]
\[
\theta_x = \frac{x}{z} = \frac{\lambda}{100 \times 10^{-6}} = \frac{0.5 \times 10^{-6}}{100 \times 10^{-6}}
\]
\[
\theta_x = 5 \mathrm{mrad} = 0.29^\circ
\]