Fourier optics

Fourier optics treats an optical field in a plane as a superposition of plane waves travelling in different directions, each corresponding to one spatial frequency of the field's transverse pattern. A pattern that varies with period $\Lambda$ across the plane is a plane wave, or a pair of them, tilted by an angle with $\sin\theta = \lambda/\Lambda$. Propagation, diffraction and imaging then become linear filtering operations, described by transfer functions and computed with Fourier transforms. The framework rests on the scalar Huygens-Fresnel description of propagation (see Huygens' principle) and is the basis of modern imaging theory, holography, optical signal processing and most numerical beam-propagation codes.

Angular spectrum and free-space propagation

A field $U(x, y, 0)$ has an angular spectrum $A(f_x, f_y)$, its two-dimensional Fourier transform. Each component propagates a distance $z$ with the transfer function

$$H(f_x, f_y) = \exp\!\left(i 2\pi z \sqrt{\lambda^{-2} - f_x^2 - f_y^2}\right).$$

Components with $f_x^2 + f_y^2 < 1/\lambda^2$ acquire only a phase. Components above $1/\lambda$ have an imaginary square root and decay exponentially as evanescent waves; at 633 nm the limit is about 1580 cycles/mm. Detail finer than about half a wavelength is therefore lost within a few wavelengths of the object, which is the physical origin of the far-field diffraction limit. At large distances the angular spectrum reduces to the Fraunhofer approximation, and the far field becomes the Fourier transform of the aperture field itself; the Fresnel number indicates which regime applies.

The lens as a Fourier transformer

A thin lens of focal length $f$ adds a quadratic phase that brings the far field in from infinity to its back focal plane. With the input field in the front focal plane, the field in the back focal plane is

$$U_f(x, y) = \frac{1}{i\lambda f}\, \tilde U\!\left(\frac{x}{\lambda f}, \frac{y}{\lambda f}\right),$$

where $\tilde U$ is the Fourier transform of the input. Position in the focal plane maps to spatial frequency by $f_x = x/(\lambda f)$. A sinusoidal grating of 100 lines/mm illuminated at 632.8 nm, with $f = 200$ mm, produces first-order spots at

$$x = \lambda f\,\nu = 12.7\ \text{mm}$$

on either side of the zero-order spot. If the input is not in the front focal plane, the same transform appears multiplied by a quadratic phase; the intensity pattern, which is what a camera records, is unchanged.

The 4f system and spatial filtering

Two lenses separated by the sum of their focal lengths form a 4f system: the first produces the Fourier transform in the shared focal plane, and the second transforms it back into an image, inverted and scaled by $f_2/f_1$. A mask placed in the Fourier plane multiplies the spectrum and so filters the image. A small pinhole passes only low frequencies and cleans a laser beam of high-angle scatter, the usual spatial filter; a central stop removes the zero order and gives dark-field images; a quarter-wave phase plate on the zero order converts phase variations into intensity, which is Zernike phase contrast. The Abbe-Porter experiments, in which a wire mesh is imaged while selected diffraction orders are blocked, demonstrate the principle directly.

Imaging as filtering

In an imaging system the pupil acts as the Fourier-plane filter. With coherent illumination the amplitude transfer function is the pupil itself, and the cutoff spatial frequency is $\mathrm{NA}/\lambda$. With incoherent illumination the system is linear in intensity; its optical transfer function is the autocorrelation of the pupil, with cutoff $2\,\mathrm{NA}/\lambda$, and its magnitude is the modulation transfer function. For an objective with NA 0.1 at 500 nm the cutoffs are 200 cycles/mm coherent and 400 cycles/mm incoherent. The inverse transform of the pupil gives the amplitude point spread function, whose squared magnitude for a circular pupil is the Airy pattern.

Numerical Fourier optics

Propagation codes compute the angular-spectrum or Fresnel integral with fast Fourier transforms. The quadratic phase in the Fresnel kernel must be sampled finely enough to avoid aliasing; for a square window of width $L$ sampled with $N$ points, the critical condition is $N = L^2/(\lambda z)$. A 10 mm window at 633 nm propagated 1 m needs about 158 points per side at that condition, and the transfer-function method is accurate for shorter distances while the impulse-response method suits longer ones. Zero-padding is needed to prevent light wrapping around the window edges.

Pitfalls

The scalar theory ignores polarization and fails for apertures comparable to the wavelength and for high-NA focusing, where vector diffraction theory is required. Paraxial Fourier relations assume small angles; at large angles the map from position to frequency is $\sin\theta$, and lenses introduce aberrations that distort the transform. A real lens aperture also truncates the spectrum, which acts as an additional low-pass filter.

Common questions

Why does a lens perform a Fourier transform?

Each plane-wave component leaving the input focuses to a single point in the back focal plane, at a position proportional to its direction. The field there is a map of the amplitude of each direction, which is the Fourier transform of the input.

What is a spatial frequency?

The number of cycles per unit length of a transverse pattern, in cycles/mm or lines/mm. It corresponds to a propagation direction with $\sin\theta = \lambda f_x$.

What is the difference between Fresnel and Fraunhofer diffraction in this framework?

Fraunhofer diffraction is the far-field limit, where the pattern is the Fourier transform of the aperture field; Fresnel diffraction retains a quadratic phase term and applies closer to the aperture.

References: J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 4; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 11; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 8.