Full text of article: Russian language.
In the 1970s, researchers were not satisfied any more with normal ways of light control via lenses, mirrors and diffraction gratings. They needed optics to solve more complicated problems. Computer-synthesized holograms, especially digital holograms, could generally carry out any optical field transformations. However, performance of a first-order diffraction hologram made it energy-inefficient. Thus, optical elements were required, which could focus a powerful laser beam into a small freeform domain with practically one hundred per cent efficiency. So, in the early 1980s, a completely new type of optical elements – focusators - appeared. Analytical solutions of practical critical problems in laser-beam focusing process were obtained within the framework of planar optics: focusators were designed and synthesized into a direct-axis segment , a transverse segment, a freeform curve segment , an O-ring , and a constant-rate circle or square . These diffractive optical elements (DOEs) for infrared and visible light spectrums were manufactured using photolithography technology, which is normal for microelectronics, and showed high diffraction efficiency (70-90%). Focusators are used as power optics tools when solving materials processing tasks.
These early Sisakyan’s and his co-authors’ works contained many fundamental ideas of diffractive optics. So, selection of an approximation type, through which focusators were calculated, was driven by maximum diffraction efficiency achieved. This required a focusator’s microrelief function to be as smooth as possible and to have as few disconnections as possible, which led to uncontrolled light-energy scattering. Therefore, within the framework of ray optics, using the eikonal equation and the differential law of energy conservation, during radiation transfer from a focusator’s plane to a target plane, proper analytical solutions were sought for some key problems with particular symmetries. Of course, in the performance of ray-optics problems, diffraction phenomena, which play an essential role in shaping a focal region, can’t be taken into consideration; that’s why geometric focusators with high efficiency form the focal intensity distribution with a considerable error (about 30%). In order to consider diffraction effects in calculating focusators, we need to decide diffracting Fresnel-Kirchhoff integral equations, using the successive approximations method. However, as turns out, an iterative process of searching a diffractive solution may faster converge when selecting a ray-optical solution, as a focusator’s initial phase approximation. In addition, during iterations, the smoothness of the original solution is insignificantly violated. If an iterative optimizing process starts with a random phase, as is done at calculating phase-only synthetic holograms, the overall DOE phase function will have a discontinuity set, and the overall focusator’s surface will look like a rough surface.
When solving a beam freeform focusing problem with a minimum diffraction thickness, while mapping a two-dimensional focusator’s plane in a one-dimensional point set, it is required the whole lines on the focusator’s surface be focused at some isolated curve points. These conditions resulted in occurrence of a focusator’s layer concept, which eventually became a basic concept of geometric focusators theory.
In paper  the authors suggest an idea of calculating a DOE phase function as a complementary aberration function, which should be added to the phase function of an ideal converging lens. Thus, the focusator in an axial segment may be represented as a lens with large spherical aberration, whereas the focusator in a transverse segment – as a spherical lens with large astigmatism.
In paper  the authors put forward the idea of calculating module (composite) focusators, which focus the light on a freeform plane curve. The whole complex curve is composed from more simple curves (a straight line and semicircle), where a particular focusator’s surface area (a single module) is focused on each simple curve.
In paper  the synthesis and testing of a reflection-mode focusator in a ring with minimum diffraction width is first reported for a CO2-laser, 10.6 microns in wavelength. DOE’s diameter equals to 2.5 mm, and the ring, 25 mm in diameter, is formed at 30 cm therefrom. The relief surface is almost even (255 relief levels are fulfilled via local washout of photosensitive emulsion and subsequent deposition of an aluminum layer).
As reported in paper , the focusator was successfully tested by General Motors (USA) on a segment with larger light intensity on the edges thereof to harden steel by a 3-kilowatt CO2 laser; the durability of a reinforced steel strip was increased five- or sevenfold.
Ever-growing use of laser light sources and light fibers in the 1970s, which generated multimode light fields, resulted in necessary development of a new class of devices – coherent light mode content analyzers. So, in the early 1980s new diffractive optical elements (modans), designed for selection of spatial laser modes, appeared. Modans (spatial filters) were designed and manufactured for selection of Gauss-Laguerre and Gauss-Hermite modes [5, 6]. If a prism or a diffraction grating spatially divides chromatic light spectral components, the modans can spatially divide transversal mode components of laser light beams. Using such diffractive optical elements, it’s possible not only to define a target mode at the output of an optical fiber or a laser resonator, but also, in contrast, to form one-mode or multi-mode laser beams for efficient introducing the light into fibers.
In paper  the authors explain how amplitude filters were first synthesized matched with particular Gauss-Laguerre modes. An alternating function, describing filter’s transmission, was recorded on an amplitude carrier using a special encoding technique with addition of constant amplitude displacement.
The first modans had size of 6.4 x 6.4 mm, 25 microns resolution, and 256 halftone values of the transmission amplitude.
In paper  the first experiment on selection of Gauss-Laguerre modes excited by Gaussian beams in graded-index optical fibers with a near-parabolic refraction index profile was described. Positive and negative parts of a real-valued function, describing the mode, were implemented in the form of different amplitude spatial filters. The first four fiber modes were analyzed.
Meantime, with the appearance of high-precision electronic or optical plotters with a resolution of less than 1 micron, many ideas contained in these early works were successfully implemented. Multiordinal phase gray-level modans (up to 64 phase levels) were synthesized, which were capable to simultaneously sort up to 25 different modes excited in waveguides with constant and graded indices.
A new direction in optics – singular optics – has been successfully developed in recent years, which studies coherent light fields around zero intensity points. Light propagation near these critical points (a phase of electromagnetic modes isn’t defined in these points) is analogous to a vortex. It’s possible to form the light field with an angular momentum using Bessel-optics elements. These diffractive optical elements were first designed and fabricated in 1984 . Bessel-optics may also be used to optically perform Hankel integral transforms and to form nondiffraction laser beams.
It is first stated in paper  that, using Bessel optics, the Guassian beam may be transferred into an annular beam, and thus there is no indicative central intensity peak presented during the light-ring formation using a conical axicon and the spherical lens. It is also argued that it’s possible, using Bessel optics (angular harmonics), to form and select higher-order Bessel modes that are excited by light fibers with the constant refractive index. In paper  it’s mentioned that threaded Bessel filters were manufactured by whitening amplitude masks.
Calculation of diffractive optical elements within the framework of the scalar diffraction Kirchhoff-Fresnel theory is based on iterative solution of integral equations. There are currently many iterative and gradient methods for synthesis of DOEs, and one of the first works on iterative calculation of multiordinal phase diffraction gratings was performed in 1986 . The iterative approach to calculating DOEs enables to naturally account for discretization and quantization of the phase function, which can’t be avoided in synthesizing DOEs in a multiple substrate etching process, using amplitude masks. Diffraction gratings with symmetrical arrangement of orders, equal in intensity, are of practical interest. To achieve high-accuracy equality of orders intensity, the scanning diffraction occurred due to discretization is considered in paper . In paper  a stability analysis of obtained solutions is compared to technology errors (location inaccuracy of different grooves, height inaccuracy of a phase step for a binary grating). The diffractive optical element represented by a multiordinal diffraction grating with the spherical lens was calculated and fabricated using an intelligence technique.
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