"Flatland" is the title of a science fiction story, written in 1880 by E.A. Abbott. The creatures of Flatland, living in their two-dimensional universe, are inspected and manipulated by 3D-people like we are. Here we show how the optics part of this science fiction story can be implemented -- for fun and profit.
The propagation law of the Wigner distribution function in the first-order non-orthogonal optical systems is described by using the linear canonical transform integral. The Wigner matrices for the usual optical components (free space, spherical and cylindrical lenses, and linear phase filter) are presented in four-dimensional phase space domain. Then with Wigner algebra, we analyze basic and more general optical configurations for performing a set of linear unitary coordinate transformations. These configurations are comprised of refractive spherical and cylindrical lenses that are readily available.
An optical spatial filtering system, designed to operate with totally incoherent light sources is presented. When correlating with such polychromatic light sources, one major problem is the dependence of the correlation response scaling on wavelength. A theoretical system that exactly compensate for this problem is analyzed. A practical configuration for implementing such a system, satisfying the theoretical solution up to first order in wavelength, is proposed. This configuration includes a combination of diffractive and refractive lenses and a gray scale filter. The possibility of higher order approximations is considered.
An achromatic Fourier system (AFS) performs optically a Fourier transformation in spatially coherent white light, without color blurring. Our design method uses matrix algebra, applied upon the Wigner distribution function. The matrix procedure is similar to what is used in geometrical objects. However, our approach is valid also for wave optics. That is important because the AFS does contain not only refractive lenses, but also diffractive lenses (Fresnel zone plates).
We present a system in which it is possible to distort the intensity distribution of an object I(x,y) into I(Ax + By,Cx + Dy), where the ABCD matrix has a unit determinant. The coefficients A, B, C and D can be varied independent of each other by rotating some cylindrical lenses. The degree of coherence is arbitrary. The number of pixels is limited in the usual way by the size of the object, by the wavelength, and by the F-number of the lenses.
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