Fourier Optics

The transmittance of a single slit centred at $x = 0$ is $t_{\mathrm{slit}(x) = \mathrm{rect}(x/a)}$. The full grating is $N$ slits:

$t(x) = \sum_{n=0}^{N-1} t_{\mathrm{slit}(x - nd) = t_{\mathrm{slit}(x) * \sum_{n=0}^{N-1} \delta(x - nd)}}$

The Fourier transform is:

$\tilde{t}(u) = \mathcal{F}\{t_{\mathrm{slit}\} \cdot \mathcal{F}\left\{\sum_{n=0}^{N-1}\delta(x - nd)\right\}}$

$= a\,\mathrm{sinc}(\pi a u) \cdot \sum_{n=0}^{N-1} e^{-2\pi i n d u} = a\,\mathrm{sinc}(\pi a u) \cdot \frac{\sin(N\pi d u)}{\sin(\pi d u)}$

The intensity is:

$I(u) = I_0\,a^2\,\mathrm{sinc}^2(\pi a u)\,\frac{\sin^2(N\pi d u)}{\sin^2(\pi d u)}$

The first factor is the single-slit envelope; the second is the $N$-slit interference pattern. Principal maxima occur at $du = m$ (integer $m$), giving the grating equation $d\sin\theta = m\lambda$.

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The aperture function is $t(r) = 1$ for $r \leq a$ and $t(r) = 0$ for $r > a$. By circular symmetry, the Fourier transform in polar coordinates is:

$\tilde{t}(q) = 2\pi\int_0^a J_0(2\pi q r)\, r\, dr$

Where $J_0$ is the Bessel function of the first kind and $q = \sin\theta/\lambda$ is the radial spatial frequency. Using the identity:

$\int_0^a J_0(2\pi q r)\, r\, dr = \frac{a}{2\pi q}J_1(2\pi q a)$

$\tilde{t}(q) = \pi a^2 \cdot \frac{2J_1(\alpha)}{\alpha}$

Where $\alpha = 2\pi a q = 2\pi a\sin\theta/\lambda$. The intensity is:

$I(\theta) = I_0\left(\frac{2J_1(\alpha)}{\alpha}\right)^2$

This is the Airy pattern. The first zero occurs at $\alpha = 3.832$Giving the angular radius of the first dark ring:

$\sin\theta_1 = 1.22\,\frac{\lambda}{d}$

Where $d = 2a$ is the diameter.

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