In what sense does a lens perform a Fourier transform?

The far-field diffraction pattern of a field is its Fourier transform, and a lens brings that far field to a finite distance, its focal plane, so the field there is the Fourier transform of the field at the front focal plane.

What is spatial filtering?

By placing a mask in the Fourier plane of a lens one can block or attenuate selected spatial frequencies, which removes noise, sharpens edges, or otherwise reshapes the image when it is reconstituted by a second lens.

Fourier Optics

Fourier optics describes the propagation and imaging of light in terms of spatial frequencies, treating a lens as an instrument that performs a Fourier transform.

Definition

The treatment of optical fields and imaging systems through their spatial-frequency content, in which propagation and lenses act as linear operations, notably Fourier transformation and filtering, on the field.

Scope

This topic covers the application of Fourier analysis to optical propagation and imaging. It includes the angular-spectrum representation of wave propagation, the result that a lens forms the Fourier transform of an input field in its back focal plane, the transfer-function description of imaging systems, the point-spread and optical-transfer functions, and spatial filtering and optical signal processing. It provides a unified, systems-level language for diffraction and imaging and underlies holography and computational imaging.

Core questions

How is the propagation of light described in terms of spatial frequencies?
Why does a lens produce the Fourier transform of a field in its focal plane?
How do the point-spread and transfer functions characterize an imaging system?
How can spatial filtering manipulate an image?

Key concepts

spatial frequency
angular spectrum
Fourier-transform plane
point-spread function
optical transfer function
spatial filtering
convolution
coherent and incoherent imaging

Key theories

Fourier-transforming property of a lens: A converging lens produces, in its back focal plane, a field proportional to the two-dimensional Fourier transform of the field in its front focal plane, making spatial frequencies physically accessible for filtering.
Linear-systems description of imaging: An imaging system acts as a linear, shift-invariant filter whose point-spread function and its transform, the optical transfer function, determine how spatial frequencies of the object are reproduced in the image.

Clinical relevance

Fourier-optics concepts underlie the optical transfer function used to specify the image quality of medical imaging optics and the computational and deconvolution methods that enhance microscope and ophthalmic images.

History

Abbe's theory of microscope imaging in the 1870s introduced the idea that an image is synthesized from the diffracted spatial-frequency components captured by the objective. This spatial-frequency viewpoint was developed into the modern discipline of Fourier optics in the mid-twentieth century, codified in Goodman's influential textbook.

Key figures

Ernst Abbe
Joseph W. Goodman
Frits Zernike

Seminal works

goodman2017
bornwolf1999

Frequently asked questions

In what sense does a lens perform a Fourier transform?: The far-field diffraction pattern of a field is its Fourier transform, and a lens brings that far field to a finite distance, its focal plane, so the field there is the Fourier transform of the field at the front focal plane.
What is spatial filtering?: By placing a mask in the Fourier plane of a lens one can block or attenuate selected spatial frequencies, which removes noise, sharpens edges, or otherwise reshapes the image when it is reconstituted by a second lens.