Fourier optics

Fourier optics is the study of classical optics using Fourier transforms, 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 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, 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 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 can be regarded as an infinite number of "plane wave modes", all of which could 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 as the conjugate of the spatial 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 emanated from the reticle is diffracted and each diffracted light may correspond to a different spatial frequency. 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 or a material medium. 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, and t represents time.

The wave equation

Fourier optics begins with the homogeneous, scalar wave equation :
where is the speed of light and u is a real-valued Cartesian component of an electromagnetic wave propagating through a free space.

Sinusoidal steady state

If light of a fixed frequency in time/wavelength/color is assumed, then, based on the engineering time convention, which assumes an time dependence in wave solutions at the angular frequency with where is a time period of the waves, the time-harmonic form of the optical field is given as
where is the imaginary unit, is the operator taking the real part of,
is the angular frequency of light waves, and
is, in general, a complex quantity, with separate amplitude in non-negative real number and phase.

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 in vacuum, is the wave number, is the spatial part of a complex-valued Cartesian component of an electromagnetic wave. Note that the propagation constant and the angular frequency are linearly related to one another, a typical characteristic of transverse electromagnetic waves in homogeneous media.
Since the originally desired real-valued solution of the scalar wave equation can be simply obtained by taking the real part of, solving the following equation, known as the Helmholtz equation, is mostly concerned as treating a complex-valued function is often much easier than treating the corresponding real-valued function.

Solving the Helmholtz equation

Solutions to the Helmholtz equation in the Cartesian coordinate system may readily be found via the principle of separation of variables for partial differential equations. This principle says that in separable orthogonal coordinates, an elementary product solution to this wave equation may be constructed of the following form:
i.e., as the product of a function of x, times a function of y, times a function of z. If this elementary product solution is substituted into the wave equation, using the scalar Laplacian in the Cartesian coordinates system
then the following equation for the 3 individual functions is obtained
which is readily rearranged into the form:
It may now be argued that each quotient in the equation above must, of necessity, be constant. To justify this, let's say that the first quotient is not a constant, and is a function of x. Since none of the other terms in the equation has any dependence on the variable x, so the first term also must not have any x-dependence; it must be a constant. This constant is denoted as -k_x². Reasoning in a similar way for the y and z quotients, three ordinary differential equations are obtained for the f_x, f_y and f_z, along with one separation condition:
Each of these 3 differential equations has the same solution form: sines, cosines or complex exponentials. We'll go with the complex exponential as to be a complex function. As a result, the elementary product solution is
with a generally complex number. This solution is the spatial part of a complex-valued Cartesian component of a propagating plane wave. is a real number here since waves in a source-free medium has been assumed so each plane wave is not decayed or amplified as it propagates in the medium. The negative sign of in a wave vector means that the wave propagation direction vector has a positive -component, while the positive sign of means a negative -component of that vector.
Product solutions to the Helmholtz equation are also readily obtained in cylindrical and spherical coordinates, yielding cylindrical and spherical harmonics.

The complete solution: the superposition integral

A general solution to the homogeneous electromagnetic wave equation at a fixed time frequency in the Cartesian coordinate system may be formed as a weighted superposition of all possible elementary plane wave solutions as
with the constraints of, each as a real number, and where. In this superposition, is the weight factor or the amplitude of the plane wave component with the wave vector where is determined in terms of and by the mentioned constraint.
Next, let
Then:
The plane wave spectrum representation of a general electromagnetic field in the equation is the basic foundation of Fourier optics, because at z = 0, the equation simply becomes a Fourier transform relationship between the field and its plane wave contents.
Thus:
and
All spatial dependence of each plane wave component is described explicitly by an exponential function. The coefficient of the exponential is a function of only two components of the wave vector for each plane wave, for example and, just as in ordinary Fourier analysis and Fourier transforms.

Connection between Fourier optics and imaging resolution

Let's consider an imaging system where the z-axis is the optical axis of the system and the object plane is the plane at. On the object plane, the spatial part of a complex-valued Cartesian component of a wave is, as shown above, with the constraints of, each as a real number, and where. The imaging is the reconstruction of a wave on the object plane on the image plane via the proper wave propagation from the object to the image planes, and the wave on the object plane, that fully follows the pattern to be imaged, is in principle, described by the unconstrained inverse Fourier transform where takes an infinite range of real numbers. It means that, for a given light frequency, only a part of the full feature of the pattern can be imaged because of the above-mentioned constraints on ; a fine feature which representation in the inverse Fourier transform requires spatial frequencies, where are transverse wave numbers satisfying, can not be fully imaged since waves with such do not exist for the given light of , and spatial frequencies with but close to so higher wave outgoing angles with respect to the optical axis, requires a high NA imaging system that is expensive and difficult to build. For, even if complex-valued longitudinal wavenumbers are allowed, give rise to light decay along the axis so waves with such may not reach the image plane that is usually sufficiently far way from the object plane.
In connection with photolithography of electronic components, these and are the reasons why light of a higher frequency or a higher NA imaging system is required to image finer features of integrated circuits on a photoresist on a wafer. As a result, machines realizing such an optical lithography have become more and more complex and expensive, significantly increasing the cost of the electronic component production.