Fourier Optics

A converging lens performs an optical Fourier transform: it converts a spatial pattern in its front focal plane into a frequency spectrum at the rear focal plane. This forms the basis of spatial filtering and holography.

Key Concepts

Far-field diffraction pattern = Fourier transform of aperture function
Lens performs optical Fourier transform with scale f·λ
Abbe theory: image formation as two-step Fourier transform
Spatial filtering: block certain frequencies in the Fourier plane
Holography records both amplitude and phase information

Key Equations

Fraunhofer diffraction

U(f_x, f_y) = \iint u(x,y)e^{-2\pi i(f_x x + f_y y)}dx\,dy

Spatial frequency

f_x = \frac{x}{\lambda f},\quad f_y = \frac{y}{\lambda f}

Rectangular aperture FT

\text{sinc}(af_x)\text{sinc}(bf_y)

Circular aperture FT

\frac{J_1(\pi D f_r)}{\pi D f_r/2}\text{ (Airy pattern)}

Worked Example

Example Problem

Problem

A rectangular aperture (a=1 mm wide) is illuminated by λ=500 nm. At a lens focal plane (f=100 mm), find the first zero of the Fourier transform.

Solution

First zero of sinc at af_x=1 → f_x=1/a. Position x = λf × f_x = λf/a = 500×10⁻⁹×100×10⁻³/10⁻³ = 0.05 mm = 50 μm.

Practice

Exercises

7 problems

1 of 7

A slit (a=0.5 mm) is illuminated by λ=600 nm. With a lens of f=200 mm, find the position of the first dark fringe in the Fourier plane in μm.

Your answer

μm

2 of 7

A diffraction-limited imaging system uses λ=500 nm and NA=0.5. Find the Abbe resolution limit in nm.

Your answer

3 of 7

A circular aperture (D=2 mm) is at a lens (f=50 cm) illuminated by λ=550 nm. Find the radius of the first Airy ring in the focal plane in μm.

Unlock Exercise 3