Fourieroptics

Fourier optics is a branch of optics that analyzes how light fields are transformed by optical systems using Fourier transform mathematics. It treats diffraction and imaging in terms of spatial frequency content, describing how an optical field at one plane is connected to fields at other planes by linear, shift-invariant operators. Under the paraxial and coherent approximation, a thin lens performs a Fourier transform between the plane at its front focal surface and the back focal surface. Consequently, the field in the lens's back focal plane is proportional to the two-dimensional Fourier transform of the field in the input plane. This makes lenses and apertures natural Fourier transformers, so that an optical system can be analyzed by its transfer function in the spatial frequency domain.

Two key concepts: the point spread function (PSF) and the optical transfer function (OTF). The PSF describes the response to a point source, while the OTF (and its magnitude, the MTF) describes how spatial frequencies are attenuated by the system. Coherent imaging preserves phase and is governed by amplitude transfer, while incoherent imaging relates to intensity via intensity transfer.

Practical implementations include the 4f correlator, where two lenses with their focal planes in between realize two successive Fourier transforms, enabling spatial filtering, pattern recognition, and optical computing. Fourier optics underpins holography, digital holography, optical filtering, and microscopy, and informs the design of imaging systems, displays, and sensors.

Limitations include finite aperture, aberrations, and nonparaxial effects; real scenes may be illuminated incoherently, requiring different models. Extensions cover vector fields, broadband illumination, and nonideal lenses.