Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts (also called phasefronts) whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.

A curved phasefront may be synthesized from an infinite number of these "natural modes" i.e., from plane wave phasefronts oriented in different directions in space. When an expanding spherical wave is far from its sources, it is locally tangent to a planar phase front (a single plane wave out of the infinite spectrum), which is transverse to the radial direction of propagation. In this case, a Fraunhofer diffraction pattern is created, which emanates from a single spherical wave phase center. In the near field, no single well-defined spherical wave phase center exists, so the wavefront isn't locally tangent to a spherical ball. In this case, a Fresnel diffraction pattern would be created, which emanates from an extended source, consisting of a distribution of (physically identifiable) spherical wave sources in space. In the near field, a full spectrum of plane waves is necessary to represent the Fresnel near-field wave, even locally. A "wide" wave moving forward (like an expanding ocean wave coming toward the shore) can be regarded as an infinite number of "plane wave modes", all of which could (when they collide with something such as a rock in the way) scatter independently of one other. These mathematical simplifications and calculations are the realm of Fourier analysis and synthesis – together, they can describe what happens when light passes through various slits, lenses or mirrors that are curved one way or the other, or is fully or partially reflected.

Fourier optics forms much of the theory behind image processing techniques, as well as applications where information needs to be extracted from optical sources such as in quantum optics. To put it in a slightly complex way, similar to the concept of frequency and time used in traditional Fourier transform theory, Fourier optics makes use of the spatial frequency domain (k_x, k_y) as the conjugate of the spatial (x, y) domain. Terms and concepts such as transform theory, spectrum, bandwidth, window functions and sampling from one-dimensional signal processing are commonly used.

Fourier optics plays an important role for high-precision optical applications such as photolithography in which a pattern on a reticle to be imaged on wafers for semiconductor chip production is so dense such that light (e.g., DUV or EUV) emanated from the reticle is diffracted and each diffracted light may correspond to a different spatial frequency (k_x, k_y). Due to generally non-uniform patterns on reticles, a simple diffraction grating analysis may not provide the details of how light is diffracted from each reticle.

Propagation of light in homogeneous, source-free media

Light can be described as a waveform propagating through a free space (vacuum) or a material medium (such as air or glass). Mathematically, a real-valued component of a vector field describing a wave is represented by a scalar wave function u that depends on both space and time:

where represents a position in a three dimensional space (in the Cartesian coordinate system here), and t represents time.

The wave equation

Fourier optics begins with the homogeneous, scalar wave equation (valid in source-free regions):

where ${\displaystyle c}$ is the speed of light and u(r,t) is a real-valued Cartesian component of an electromagnetic wave propagating through a free space (e.g., u(r, t) = E_i(r, t) for i = x, y, or z where E_i is the i-axis component of an electric field E in the Cartesian coordinate system).

Sinusoidal steady state

If light of a fixed frequency in time/wavelength/color (as from a single-mode laser) is assumed, then, based on the engineering time convention, which assumes an ${\displaystyle e^{i\omega t}}$ time dependence in wave solutions at the angular frequency ${\displaystyle \omega =2\pi f}$ with ${\displaystyle f=1/\tau }$ where ${\displaystyle \tau }$ is a time period of the waves, the time-harmonic form of the optical field is given as

where ${\displaystyle i}$ is the imaginary unit, ${\displaystyle \operatorname {Re} \left\{x\right\}}$ is the operator taking the real part of ${\displaystyle x}$, is the angular frequency (in radians per unit time) of light waves, and is, in general, a complex quantity, with separate amplitude ${\displaystyle a}$ in non-negative real number and phase ${\displaystyle \phi }$.

The Helmholtz equation

Substituting this expression into the scalar wave equation above yields the time-independent form of the wave equation,

where with the wavelength ${\displaystyle \lambda }$ in vacuum, is the wave number (also called propagation constant), $$Zdroj:https://en.wikipedia.org?pojem=Fourier_optics
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