Fourier Optics

In the Fraunhofer (far-field) limit, the diffraction pattern of an aperture with transmission function $t(x, y)$ is the Fourier transform:

$U(x", y") = \frac{e^{ikz}}{i\lambda z}\,e^{ik(x'^2 + y'^2)/(2z)}\iint t(x, y)\,e^{-ik(xx' + yy')/z}\,dx\,dy$

Where $(x', y')$ are coordinates in the observation plane at distance $z$ from the aperture.

Defining spatial frequencies $f_x = x'/(\lambda z)$, $f_y = y'/(\lambda z)$:

$U(f_x, f_y) \propto \mathcal{F}\{t(x,y)\}(f_x, f_y)$

This correspondence between diffraction and Fourier transforms is the foundation of Fourier optics and has profound implications for image processing and optical information processing.

Ernst Abbe (1873) showed that a microscope forms an image by taking two Fourier transforms: the objective lens performs the first Fourier transform (creating the diffraction pattern at its back focal plane), and the eyepiece (or tube lens) performs the inverse transform.

Resolution limit: The finest spatial frequency that can pass through the objective is:

f_{\max} = \frac{\text{NA}{\lambda}}

Where $\text{NA} = n\sin\theta$ is the numerical aperture. The minimum resolvable distance (Abbe limit):

$d_{\min} = \frac{\lambda}{2\,\text{NA}}$

For green light ($\lambda = 550$ nm) and NA = 1.4 (oil immersion): $d_{\min} \approx 196$ nm.

Since the back focal plane of a lens contains the spatial frequency spectrum of the input, placing a mask (spatial filter) in this plane modifies the image:

• Low-pass filter: Blocks high spatial frequencies $\to$ smooths the image, removes fine detail
• High-pass filter: Blocks low frequencies $\to$ enhances edges, removes uniform background
• Phase contrast microscopy: (Zernike, 1942) Adds a $\pi/2$ phase shift to the undiffracted (DC) component, converting phase variations into intensity variations. This makes transparent biological specimens visible without staining.

• Confusing Fraunhofer and Fresnel diffraction. Fraunhofer diffraction applies in the far field ($z \gg a^2/\lambda$ where $a$ is the aperture size). Fresnel diffraction (near field) uses a quadratic phase factor and is not simply a Fourier transform.
• Forgetting the quadratic phase factor in the Fresnel regime. The Fraunhofer integral approximation drops the quadratic phase term, but it is present in the full Fresnel diffraction integral.
• Assuming the Abbe limit is the only resolution limit. The Abbe limit assumes coherent illumination; the Rayleigh criterion (for resolved point sources) gives $d = 0.61\lambda/\text{NA}$, slightly different.
• Thinking spatial filtering only removes noise. Spatial filtering can also introduce artifacts (ringing from hard-edged low-pass filters, edge enhancement exaggeration from high-pass filters).

Problem 1. A 4f imaging system has a lens with focal length $f = 50$ cm and aperture diameter $D = 2$ cm. The input is illuminated with $\lambda = 633$ nm. What is the cutoff spatial frequency?

Solution. The cutoff frequency is determined by the lens aperture acting as a low-pass filter. The maximum spatial frequency that passes through the system is $f_{\max} = D/(2\lambda f)$: $f_{\max} = 0.02/(2 \times 633\times 10^{-9} \times 0.5) = 31,600$ cycles/m. Finer details in the input are blocked, giving a minimum feature size of $1/f_{\max} \approx 31.6\ \mu$m. $\blacksquare$

Problem 2. Derive the point spread function (PSF) of a circular aperture of diameter $D$.

Solution. The amplitude PSF is the Fourier transform of the aperture function (a circle of diameter $D$). This gives the Airy pattern: $I(r) = I_0[2J_1(kDr/(2f))/(kDr/(2f))]^2$, where $J_1$ is the Bessel function of the first kind. The first zero occurs at $r = 1.22\lambda f/D$, which is the Rayleigh criterion. $\blacksquare$

• Microscopy: Structured illumination microscopy (SIM) uses patterned illumination to encode high-frequency information, doubling resolution beyond the Abbe limit. Stimulated emission depletion (STED) microscopy breaks the diffraction barrier entirely.
• Holography: Digital holography records the full complex field (amplitude and phase) using Fourier optics principles, enabling numerical refocusing and 3D imaging.
• Astronomy: Adaptive optics corrects wavefront distortions in real time using Fourier optics concepts. Aperture synthesis in radio astronomy reconstructs images from sparse Fourier samples.
• Optical computing: 4f correlators perform convolution operations optically at the speed of light, used in pattern recognition and optical neural networks.