Fourier Optics

7.1 Fraunhofer Diffraction as a Fourier Transform

In the Fraunhofer limit, the diffraction pattern is the Fourier transform of the aperture function:

$E(\theta_x, \theta_y) \propto \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} t(x,y)\, e^{-i(k_x x + k_y y)}\,dx\,dy$

Where $t(x, y)$ is the transmission function of the aperture, and $k_x = k\sin\theta_x$ $k_y = k\sin\theta_y$.

A lens placed one focal length after the aperture produces the Fraunhofer pattern at its back focal Plane, performing an optical Fourier transform.

7.2 The Convolution Theorem

If the aperture is a product $t(x, y) = t_1(x, y) \cdot t_2(x, y)$The diffraction pattern is the Convolution of their individual transforms:

$\mathcal{F}\{t_1 \cdot t_2\} = \mathcal{F}\{t_1\} * \mathcal{F}\{t_2\}$

Where $*$ denotes convolution. This explains, for example, why the double-slit pattern with finite Slit width is the product of a sinc function (single slit) and a cosine-squared (double slit).

7.3 The Abbe Theory of Imaging

Abbe showed that a lens images by collecting the diffracted orders and recombining them. The Resolution limit arises because high spatial frequencies (large diffraction angles) are lost if they Fall outside the lens aperture.

The minimum resolvable spatial frequency is:

f_{\mathrm{max} = \frac{2\mathrm{NA}{\lambda}}}

Where $\mathrm{NA} = n\sin\theta_{\mathrm{max}}$ is the numerical aperture.