Fourier Optics

The Fraunhofer diffraction pattern of an aperture with transmittance function $t(x, y)$ illuminated by a plane wave is proportional to the 2D Fourier transform of the aperture function:

$E(u, v) \propto \iint t(x,y)\, e^{-2\pi i(ux + vy)}\, dx\, dy = \mathcal{F}\{t(x,y)\}(u, v)$

Where $u = \sin\theta_x/\lambda$ and $v = \sin\theta_y/\lambda$ are the spatial frequencies.

Theorem 10.1. The intensity in the Fraunhofer diffraction pattern is

$I(u,v) = I_0\,|\tilde{t}(u,v)|^2$

Where $\tilde{t}(u,v) = \mathcal{F}\{t(x,y)\}(u,v)$ is the Fourier transform of the aperture function.

Proof. The Huygens-Fresnel principle in the far field gives:

$E(x",y') = \frac{e^{ikr}}{i\lambda r}\iint t(x,y)\, e^{-ik(xx' + yy')/r}\, dx\, dy$

In the far field, $r \approx D$ and the phase factor $e^{-ik(xx' + yy')/r}$ is exactly the kernel of the Fourier transform. $\blacksquare$

Theorem 10.2 (Convolution theorem). If an aperture function is the convolution $t = t_1 * t_2$The diffraction pattern is the product of the individual diffraction patterns:

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

Corollary. If an aperture is the product $t = t_1 \cdot t_2$The diffraction pattern is the convolution of the individual patterns:

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

Problem. Use the Fourier transform to derive the intensity pattern of a grating with $N$ slits of width $a$ and spacing $d$.

Problem. Compute the Fraunhofer diffraction pattern of a circular aperture of radius $a$.

• Rayleigh criterion: Two point sources are just resolved when the centre of the Airy disc of one coincides with the first dark ring of the other, giving the minimum resolvable angle $\theta_{\min} = 1.22\lambda/d$.
• Fourier scaling property: If $t(x,y)$ is scaled by $a$, i.e., $t(x/a, y/a)$, then $\tilde{t}(u,v)$ scales as $|a|^2\tilde{t}(au, av)$. A larger aperture produces a narrower diffraction pattern.
• Parseval’s theorem: $\iint |t(x,y)|^2\,dx\,dy = \iint |\tilde{t}(u,v)|^2\,du\,dv$. The total power in the aperture equals the total power in the diffraction pattern.
• Uncertainty principle analogy: A narrow aperture (small $\Delta x$) produces a wide diffraction pattern (large $\Delta u$), and vice versa. Quantitatively, $\Delta x \cdot \Delta u \gtrsim 1$.

• Confusing Fraunhofer with Fresnel diffraction: Fraunhofer diffraction requires the far-field condition $D \gg a^2/\lambda$. At shorter distances, Fresnel (near-field) diffraction must be used, and the pattern is not a simple Fourier transform.
• Forgetting the intensity is the squared modulus: The diffraction pattern is $I(u,v) = I_0|\tilde{t}(u,v)|^2$, not $\tilde{t}(u,v)$. Phase information is lost in the intensity measurement.
• Neglecting the obliquity factor: The Huygens-Fresnel principle includes a directional cosine factor. For small angles this is approximately constant, but at large angles it modifies the pattern.
• Assuming the Fourier transform of a real function is real: Even if $t(x,y)$ is real and non-negative, $\tilde{t}(u,v)$ is generally complex. The phase of $\tilde{t}$ carries information about the spatial structure of the aperture.

• Telescope resolution: The Airy pattern sets the fundamental resolution limit of any circular-aperture optical system. The diameter of the primary mirror determines the smallest detail that can be resolved.
• Spectrometre design: A diffraction grating disperses light according to the grating equation $d\sin\theta = m\lambda$. The resolving power $R = mN$ depends on the order $m$ and the number of illuminated slits $N$.
• Spatial filtering: By placing masks in the Fourier plane (at the focal length of a lens), specific spatial frequencies can be blocked or attenuated. This enables edge enhancement, noise removal, and pattern recognition.
• Holography: A hologram records both the amplitude and phase of the diffracted field. Reconstruction involves illuminating the hologram, which acts as a complex transmittance function whose Fourier transform reproduces the original wavefront.

Problem. Use the convolution theorem to derive the double-slit diffraction pattern.

The aperture is the product of a double-slit function $t_1(x) = \mathrm{rect}((x - d/2)/a) + \mathrm{rect}((x + d/2)/a)$ and a wide rectangular window. However, it is simpler to view the double slit as a single slit convolved with two delta functions:

$t(x) = \mathrm{rect}(x/a) \cdot [\delta(x - d/2) + \delta(x + d/2)]$

Wait, the double slit is a product (two slits cut from an opaque screen). The transmittance is:

$t(x) = [\mathrm{rect}((x - d/2)/a) + \mathrm{rect}((x + d/2)/a)]$

The Fourier transform is:

$\tilde{t}(u) = a\,\mathrm{sinc}(\pi a u)\,e^{-i\pi d u} + a\,\mathrm{sinc}(\pi a u)\,e^{i\pi d u} = 2a\,\mathrm{sinc}(\pi a u)\cos(\pi d u)$

The intensity is:

$I(u) = 4I_0 a^2\,\mathrm{sinc}^2(\pi a u)\cos^2(\pi d u)$

The $\cos^2$ factor produces the double-slit interference fringes with spacing $\Delta u = 1/d$, and the sinc$^2$ factor provides the single-slit envelope.

$\blacksquare$

Problem. Find the Fraunhofer diffraction pattern of a rectangular aperture of width $a$ and height $b$.

The aperture function is separable: $t(x,y) = \mathrm{rect}(x/a)\,\mathrm{rect}(y/b)$. By the separability of the 2D Fourier transform:

$\tilde{t}(u,v) = \mathcal{F}\{\mathrm{rect}(x/a)\}(u) \cdot \mathcal{F}\{\mathrm{rect}(y/b)\}(v) = ab\,\mathrm{sinc}(\pi a u)\,\mathrm{sinc}(\pi b v)$

The intensity is:

$I(u,v) = I_0\,a^2 b^2\,\mathrm{sinc}^2(\pi a u)\,\mathrm{sinc}^2(\pi b v)$

The pattern is a product of two sinc$^2$ functions. The first zero along $u$ occurs at $u = 1/a$ (angular position $\sin\theta_x = \lambda/a$), and along $v$ at $v = 1/b$ ($\sin\theta_y = \lambda/b$). A wider aperture produces a narrower diffraction pattern in that direction.