Fourier optics
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.
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 (kx, ky).
Reasoning in a similar way for the y and z quotients, three ordinary differential equations are obtained for the fx, fy and fz, along with one separation condition:
in the Cartesian coordinate system may be formed as a weighted superposition of all possible elementary plane wave solutions as with the constraints of
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
so higher wave outgoing angles with respect to the optical axis, requires a high NA (Numerical Aperture) imaging system that is expensive and difficult to build.
In connection with photolithography of electronic components, these (1) and (2) are the reasons why light of a higher frequency (smaller wavelength, thus larger magnitude of
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.
A solution to the Helmholtz equation as the spatial part of a complex-valued Cartesian component of a single frequency wave is assumed to take the form:
radial dependence is a spherical wave - both in magnitude and phase - whose local amplitude is the FT of the source plane distribution at that far field angle.
Note that the term "far field" usually means we're talking about a converging or diverging spherical wave with a pretty well defined phase center.
The connection between spatial and angular bandwidth in the far field is essential in understanding the low pass filtering property of thin lenses.
As a side note, electromagnetics scientists have devised an alternative means to calculate an electric field in a far zone which does not involve stationary phase integration.
A DC (Direct Current) electrical signal is constant and has no oscillations; a plane wave propagating parallel to the optic (
an Infinite homogeneous media admits the rectangular, circular and spherical harmonic solutions to the Helmholtz equation, depending on the coordinate system under consideration.
The notion of k-space is central to many disciplines in engineering and physics, especially in the study of periodic volumes, such as in crystallography and the band theory of semiconductor materials.
Also, the impulse response (in either time or frequency domains) usually yields insight to relevant figures of merit of the system.
In the case of most lenses, the point spread function (PSF) is a pretty common figure of merit for evaluation purposes.
It is at this stage of understanding that the previous background on the plane wave spectrum becomes invaluable to the conceptualization of Fourier optical systems.
Concepts of Fourier optics are used to reconstruct the phase of light intensity in the spatial frequency plane (see adaptive-additive algorithm).
and the sum of the two path lengths is f (1 + θ2/2 + 1 − θ2/2) = 2f; i.e., it is a constant value, independent of tilt angle, θ, for paraxial plane waves.
No electronic computer can compete with these kinds of numbers or perhaps ever hope to, although supercomputers may actually prove faster than optics, as improbable as that may seem.
(for all kx, ky within the spatial bandwidth of the image, so that kz is nearly equal to k), the paraxial approximation is not terribly limiting in practice.
This step truncation can introduce inaccuracies in both theoretical calculations and measured values of the plane wave coefficients on the RHS of eqn.
This source of error is known as Gibbs phenomenon and it may be mitigated by simply ensuring that all significant content lies near the center of the transparency, or through the use of window functions which smoothly taper the field to zero at the frame boundaries.
In this regard, the far-field criterion is loosely defined as: Range = 2D2/λ where D is the maximum linear extent of the optical sources and λ is the wavelength (Scott [1998]).
(2.1) - the full plane wave spectrum - accurately represents the field incident on the lens from that larger, extended source.
Ragnarsson proposed a method to realize Wiener restoration filters optically by holographic technique like setup shown in the figure.
Depending on the operator and the dimensionality (and shape, and boundary conditions) of its domain, many different types of functional decompositions are, in principle, possible.
