On-axis computer-generated hologram optical elements: History of development and use. Part I
When following history of on-axis CGHOEs development, it is possible to split it into three relative stages. During the first stage basic principles of wave fields’ transformation with help of on-axis diffracting structures were defined and experimentally verified (A. Fresnel, 1816; D. Rayleigh, 1871; Ch. Soret, 1875).
The second stage is, as a whole, associated with R. Wood’s pioneer works . He, actually, made a first relief-phase binary hologram with a high diffraction efficiency in a thin gelatinous layer applied to a glass substrate. Layer’s thickness (after its exposing and developing) ensured a skew by π value between adjacent Fresnel zones.
During a third stage new technologies of obtaining CGHOEs were developed. They ensured wide practical use of these elements in optical technologies and optoelectronic instrument making. The most important accomplishment of this stage was the demonstration of possibilities, definition of practicability and justification of preferential use of on-axis CGHOEs. Elements started to get used for non-spherical surfaces control and adjustment of centered optical systems, such as, two-mirror telescopes, including space-based ones. In initial native and foreign works various ways of element synthesis were studied. Works [2, 3] describe variants of application of on-axis computer-generated holograms, work  – of off-axis ones, synthesized using the method of A.W. Lohmann, including description of CGHOEs, created using photolithography method with photographic reduction of estimated holographic field and presentation of research results of kinoform demonstration element samples . Pioneer works which demonstrated a possibility and justified the use of computer-generated holograms to control non-spherical surfaces (NSS) include both works on off-axis holography  and on on-axis holograms [2, 5–8].
It is worth mentioning that estimation of off-axis holograms took considerably more (by several orders) computing time using electronic computers, which was quite expensive back then, than estimation of according on-axis holograms.
Further development history of diffracting optics confirmed apparent advantages of on-axis CGHOEs, which include simplicity of estimation, production and certification. On-axis CGHOEs proved to be more preferable for practical optical and optoelectronic instrument making. In present time only on-axis CGHOEs are used as standard optical elements and optical compensators.
Estimation of on-axis CGHOEs is based on the idea of a "diffracted" ray. We assume the Malus-Dupin theorem and an inverse theorem of Levi-Civita formulated for reflection and refraction of light are also true for diffracted rays . It is appropriate to summarize both these theorems as: "Any two geometrical wavefronts (any two wave surfaces) can be converted into one another using one reflection, refraction or diffraction". However, non- fulfillment of a tautochronism condition in this case assumes obvious difficulties in realizing the principle of Huygens-Fresnel. Primarily, it applies to the "wavefront" definition which, in this case, should be interpreted only as a surface orthogonal to diffracted rays ("geometrical" wavefront). "Payment" for this are high requirements for monochromaticity of the used emanation source. The goal of CGHOEs estimation is determining coordinates of interference figure bands (rings) which would form on a hologram’s surface as a result of a superposition of an object and a reference wave which are defined analytically [9, 10]. To estimate computer-generated holograms a following expression is used:
Δlm (ρ) = λ (m ± 1/2Q), (1)
where Δlm (ρ) – difference between optical paths of the reference wave and the object wave at the edge of the m-th interference band, λ – working wavelength, Q – on-off time ratio (correlation between the period of iteration and the width of the band (groove) of the displayed interference figure). When estimating a computer-generated hologram edge coordinates ρ±m of each m-th interference wave must be determined.
At the early sixties of the XX century the State Institute of Applied Optics (GIPO), at that time being Kazan department of the S.I. Vavilov State Optics Institute, started research supervised by Kamil Sabirovich Mustafin on devising a technology of obtaining CGHOEs. The goal was in solving a problem of non-spherical optics control and a series of tasks on large-dimension optical instruments making based on the research. An important scientific and methodological role in formation and development of this aspect of CGHOEs was played by RAN academician Yuri Nikolayevich Denisyuk.
It should be mentioned that GIPO initially tested two methods of manufacturing of on-axis CGHOEs – using a focused light beam, and using a "cutter". Practical implementation showed that the "cutting tool" method of CGHOEs manufacturing has a number of considerable advantages (in terms of spatial frequency, size etc.). A considerable factor, which positively influenced the method’s development was availability of a closed technological cycle of ruled diffraction gratings production in GIPO, with own equipment for diamond cutter sharpening, vacuum sputtering of thin metallic reflecting layers and with experience of operating conventionsl ruling engines.
The first computer-generated hologram in GIPO was produced in 1969 on a lathe with a triangular cutting tool made of pobedit . Estimated width of ring-shaped Fresnel zones in this case was ensured by proper "dipping" of the cutting tool into aluminum substrate, flat working surface of which was previously formed using the same lathe. Its maximum spatial frequency was 30 mm-1, diffraction efficiency ~3% (amplitude hologram). Estimation of the hologram’s structure was, at the time, performed manually using "Mercedes" calculating machine.
Production of the basic technological equipment for CGHOEs manufacturing in GIPO started in late 1960s with developing of a prototype of a diamond tool-based ruling device. The prototype had a horizontal spindle rotation axis for CGHOE substrate fixing (Fig.1). A cutting tool chuck with help of a plate-like spring and an electromagnet ensured ruling of annular grooves to aluminized work surface of CGHOE substrate. In this prototype and in following generations of circular ruling engines a principle of a "bearing" spatial frequency has been implemented. That is, every estimated non-transparent (non-reflective) zone of a hologram was represented by a group of "elementary" non-transparent zones with a permanent pitch; a "bearing" spatial frequency was created – an equivalent of the "bearing" frequency in radiophysics (Fig.2) .
Further development of these technologies ensured the possibility of obtaining relief phased binary CGHOEs with diffraction efficiency up to 40% by removing these "elementary" zones .
In mid 1970s GIPO and Leningrad optomechanical association (LOMO) jointly planned and manufactured a circular ruling engine MDA-9 for cutting CGHOEs with up to 230 mm diameter (Fig.3), protected by an Invention Certificate for a "Ruling engine" invention . In MDA-9 device’s structure the spindle rotation axis for CGHOE substrate fixation was vertically oriented. Circular ruling engine MDA-9 was controlled by an electronic logical unit with a start/stop reader element; paper punched tape with data regarding coordinates of diamond cutter lowering and lifting was used as a data carrier.
In late 1990’s a circular ruling engine MDA-10 was created in GIPO based on using a piezoelectric longitudinal blank carriage drive (Fig.4). Control of CGHOE structure cutting process was performed with a PC. MDA-10 device allows manufacturing of circular CGHOEs with up to 230 mm diameter and spatial frequency up to 2000 mm-1 on substrates with flat and convex work surfaces, and cylindrical CGHOEs with dimensions of up to 70×100 mm. Inaccuracy of a specified wave surface forming does not exceed 0.05λ where wavelength λ = 633 nm with maximum spatial frequency of 100 mm-1.
Earlier, at the beginning of the 1990s, a circular ruling engine MDG-500 was developed in GIPO for cutting CGHOEs with up to 500 mm diameter, two units of which were manufactured at "Arsenal" (Kiev) plant. One of these devices (Fig.5) was delivered to GIPO in 1993. It is currently undergoing renovation for CGHOEs manufacturing with diameter up to 600 mm.
Also, methods of optical quality control of on-axis CGHOEs [9, 10] were developed based on the use of:
•measuring microscope and control rings;
•additional (certifying) computer-generated hologram with consideration of substrate and groove structure ruling inaccuracies.
It was necessary to provide local optical manufacturers with possibilities to produce optical components with non-spherical surfaces. Due to that, research was being conducted in GIPO for several decades on creating methods and means of control of such surfaces. As a result, a series of basic optical schemes based on CGHOEs use was developed for quality control of optical non-spherical surfaces during their forming, and, also, quality control of lens-based optical systems on intermediate and final stages of their manufacturing (Fig.6).
Some of them were used for controlling non-spherical surfaces on optic industry enterprises: Kazan Opto-Mechanical Plant (Kazan), Novosibirsk Instrument Making Plant (Novosibirsk), Arsenal Plant (in Kiev), Lytkaryno Optic Glass Plant (Lytkaryno, Moscow Region), Leningrad optomechanical association (St. Petersburg), Optoelectronic Instruments Complex Testing Research Institute (Sosnovyy Bor, Leningrad Region) and in the S.I.Vavilov’s State Optics Institute (St. Petersburg). To practically implement these control schemes special devices were developed in GIPO – holographic aspherometers of AG-2, AG-3, AG-4 and ASG types; several units were manufactured and supplied to various enterprises of local optic industry. Based on these devices, IFK-451 and IFK-454 instruments were developed at "PHOTON" Central Design Office (Kazan) for mass production, which perform functions of aforementioned AG and ASG type devices. IFK-451 holographic aspherometer is transportable and is used for contactless control of second- and top-order optical surfaces with diameters up to 12,000 mm for concave and up to 500 mm for convex optical components with interferometric precision. Universal holographic device IFK-454 is stationary and is designed for control:
•of form of non-spherical surfaces of any order with light diameter up to 150 mm, non-sphericity up to 3000 µm, non-sphericity gradient up to 30 µm/mm, control inaccuracy 0.03 µm;
•of form and measurement of curve radius of spherical and cylindrical surfaces with light diameter up to 200 mm and curve radius range of ±100 to ±100,000 mm.
Practical examples are presented as interferograms and CGHOE components with their summarized specifications (Fig.7-9). The fact that, in 1981 at Lytkaryno Optic Glass Plant GIPO specialists conducted control of concave parabolic mirror with 2.6 m diameter based on on-axis CGHOE developed by GIPO, proves success of holographic optical elements application in production. But these examples do not restrict the range of tasks which can be solved using on-axis CGHOEs. Apart from them this class of holographic optical elements is also used for lens decentralization value control , for estimating curve radius r of spherical and cylindrical surfaces of optical components, as kinoforms for gas flow visualizing in wind tunnels and ballistic tracks, for "expansion" and "compression" of a work spectrum region.
Inaccuracy of lens decentralization value control doesn’t exceed 10 µm. In case of spherical surface curve control of a sample glass  using CGHOE (curve radius measuring range r – ±250 to ±100,000 mm), allowable inaccuracy is ensured according to precision class 1 (GOST 2786-82 "Sample glass for spherical optical surfaces form and radius verification. Technical conditions"). This allows to estimate maximum deviation of curve radii with ±0.02% value within the range of their rated values of at least 250 to 1000 mm (inclusive) and with the value of (± 0.02r/1000)% for curve radii larger than 1000 mm. When measuring curve radii of cylindrical surfaces of optical components, namely, of cylindrical mirrors, using one-dimensional CGHOEs [9, 10] the range of curve radii alteration and inaccuracy of their measuring are similar to values which are typical when controlling spherical surfaces.
CGHOEs in form of kinoforms (with transition from binary groove form with diffraction efficiency of 35 to 40% to multi-level form with diffraction efficiency of 85 to 90%, and, within limits, to serrated form with limited diffraction efficiency of 95% ) are an alternative means of correction of chromatic aberrations of lens centered systems (objectives). This is especially valuable for UV and IR spectrums due to a limited list of materials which are transparent within these spectrums.
We must stress that this unique possibility of diffracting structures’ use was initially (1957) pointed out and justified by G.Slyusarev .
Use of diffracting structures as basic objectives or compensators in the object branch of interference instrument holographic systems allows visualizing of gas flow in wind tunnels and ballistic tracks . Inclusion of holographic objectives into these systems gives them new properties, improves technical parameters of modern opto-physical measurement systems: it increases diameter of studied gas flow from 230 to 100 mm and ensures maximum values of relative apertures reached in collimator objectives (from 1:3,5 to 1:2 and even to 1:1).
Portable diffracting element kits provide visual demonstration of separate aberrations for training purposes. For example, a training and research set of CGHOE-replicas  consists of a "non-aberrant" synthesized holographic lens (SHL), cylindrical SHL, SHL-tor, synthesized holographic simulator (SHS) of spherical aberrations of orders 3 and 5, SHS of spherical aberrations of 5-th order, SHS of sinusoidal aberrations (for all elements of this set the light diameter is 35 mm, work wavelength is 633 nm and diffraction efficiency is ~40%).
As a summary of diffracting elements’ use for "expansion" and "compression" of a work spectrum region [9, 17] we will remind about its main point. Method of spectral selection regulation of relief phase CGHOEs (alignment of diffraction efficiency relation to wavelength within the whole work spectrum range) is in selecting two transparent materials with certain difference of their refraction’s relation to wavelength. This method is implemented with the help of:
•use of sinusoidal relief phase modulation, when a holographic element consists of two phase holograms 1 and 2 with anti-phase structures, which are connected via contrary work surfaces and made using materials with different refraction values n1 and n2 (Fig. 10 a);
•selection of a system of two binary holograms 1 and 2, connected using transparent adhesive material 3 with refraction value n3 (Fig. 10 b);
•use of a single hologram 2, connected using transparent medium 1 with refraction value n1 to a protective plate 6 with refraction value n4 (Fig. 10 c).
Phase difference between two rays 4 and 5, which pass at the distance of a half of a period of a relief structure from each other can be presented as
Δφ = ( 2 π / λ ) h [ n2 ( λ ) – n1 ( λ ) ]. (2)
Maximum value of diffraction efficiency η( λ ) = ηmax is reached in case of phase difference Δφopt, the value of which depends on the form of a micro relief. Thus, for example, in case of a sinusoidal form (Fig. 10a) Δφ=1.85 rad, at the same time, maximum effectiveness in the 1-st order of diffraction is ~34%. For a graded bimodal relief type and an on-off ratio of 2 (see Fig. 10b, 10c), Δφ=1.57 rad and ηmax = 40.5%.
As it results from the proportion presented above (2): to widen the work region of the CGHOE spectrum, it is necessary to weaken (ideally – to eliminate) dependence of phase difference Δφ from wavelength λ. This is achieved by using such pair of optical materials for which dependence (3) of their refraction value difference from the wavelength changes proportionally to the wavelength in the work spectrum range
n2 ( λ ) – n1 ( λ) = a1 λ + b1. (3)
where a1 and b1 are parameters defined by optical characteristics of chosen materials, in which connection, it is advisable to have a1λ >> b1 within the work region of the spectrum. The height of CGHOE relief h is chosen from the abovementioned proportion (2) Δφ( λ) = Δφopt( λ0), where λ0 is a wavelength with which hmax is reached.
In case one hologram is made using solid-state material, the second one can be made by filling the relief structure of the first hologram with a transparent liquid material with subsequent solidification, for example, by polymerization. Non-solidifying material can also be used, in that case it must be sealed by a transparent lamina 6 (Fig.10 с). Both these holograms, which form CGHOE, are made from solid-state optical materials and can be combined using transparent adhesive.
To implement the presented method of CGHOE spectral work region widening it is necessary to provide the possibility of estimated relief phase holographic structure formation in various optical materials with a wide range of refraction and dispersion values.
As an example we will compare estimated normalized dependence of diffraction efficiency from wavelength for a CGHOE made from a pair of optical crystals "fluoric lithium – fluoric barium" and for a "reference" sample of a CGHOE made from only one material – fluoric barium. These optical materials have a good transmission in far infrared spectrum region. With optimal wavelength of 4 µm for a composite CGHOE, relief depth is 18.6 µm and for fluoric barium CGHOE relief depth is 4.4 µm. Estimations show that spectrum work region of a composite hologram stretches from 3.1 to 10 µm; "singe" CGHOE, on the other hand, has this region 7 times narrower – from 3.5 to 4.5 µm. It shows that there is a considerable widening of a spectrum work region of a CGHOE, intended for transmission, in case of proper optical material pair selection.
In some cases high spectral selectiveness of CGHOEs can be used quite efficiently. Thus, for example, in focusing systems which work in a limited spectral range in conditions of a high level of background illumination (in particular, solar-blind optical systems in wavelength regions of 250-350 nm), this property of holograms allows for a considerable increase of a "signal/background" ratio at the input of a photo receptor.
"Compression" of a spectrum work region of CGHOEs can be done the following way :
•selection of a pair of optical materials the difference of refraction values of which decreases with wavelength increase; at the same time, the faster it decreases – the better;
•realization of graded CGHOEs diffraction efficiency dependence from the phase modulation depth by means of a considerable "dosed" micro relief depth increase.
For example, we will analyze the flock of spectral curves of diffraction efficiency of a level 8 transmitting CGHOE, estimated within wavelength range from 400 to 900 nm for different phase relief depth values H: from H* = λ0 to H* = 33λ0, where λ0 = 632.8 nm; H* = H(n – 1); n = 1.5146. This flock of curves is characterized by the fact, that, with wavelength alteration, diffraction efficiency passes a row of alternating maximums and minimums, width of which sequentially decreases with decrease of wavelength. Due to that, if there is a possibility of forming of sufficiently "deep" microstructures, then, for the specified wavelength a multiple half-width decrease can be achieved. In the current example – approximately by an order of magnitude. At the same time, however, it is necessary to ensure suppression of "parasitic" secondary transmission maxima to the left and to the right of the specified wavelength. Thus, the CGHOE in this case can perform two functions at the same time: of an optical corrector (or a focuser) and of a band spectral filter.
It is interesting to study terminology evolution in the area of on-axis CGHOEs: Fresnel zone plate (Soret plate), Rayleigh zone plate, Wood relief phase binary zone plate, Gabor hologram, Slyusarev phase zone plates (the plate has a serrated profile of Frenel zones, estimation of which is performed in combination with light composite lens forming method and allows to create a single image of an object in a monochromatic light ) and Tudorovskiy plates (the plate is a "graded lens", a combination of a Slyusarev plate and a conventional thin lens with a certain optical force ), artificial hologram (this term has been proposed by GIPO engineers ; it was used in scientific and technological works until GOST 24865.1 – 81 "Holography and holographic methods of quality control. Terms and definitions" came into force), machine-made hologram, synthetic hologram, digital hologram, computer hologram, computer-generated hologram (this term, which replaced the term "artificial hologram" in 1981, was also proposed by GIPO engineers) and kinoform hologram [9, 18, 19]. Currently, a term "computer-generated hologram optical element" (CGHOE) gained a foothold in native scientific and technological literature which corresponds to its English counterpart – "computer-generated hologram optical element" (CGHOE).
Russian holographic schools: in Leningrad (St. Petersburg) – Y.N.Denisyuk and, nearly from the start – M.A. Gun with colleagues, primarily – A.F.Perveyev; in Kazan – K.S.Mustafin, A.F.Belozyorov and A.V.Lukin with colleagues, primarily – R.A.Rafikov and N.P.Larionov; in Kuybyshev (Samara) – V.A.Soyfer with colleagues; In Penza – G.I.Greysuch with colleagues; in Novosibirsk – V.P.Koronkevich and, practically from the beginning, A.G.Poleschuk with colleagues, have carried out fundamental works on estimation principles, manufacturing issues, certification and practical use of on-axis CGHOEs (kinoforms) and still continue to develop these schools for the sake of development of various branches of scientific, technological and educational fields [9, 10, 15, 18–20].
It is worth mentioning of a complex of scientific and technological works with a wide practical use carried out in the Institute of Automation and Electrometry of Siberian Branch of RAN (Novosibirsk) by V.P. Koronkevich and A.G. Poleschuk with colleagues. A distinctive feature of this CGHOE manufacturing technology is use of computer-controlled focused laser beam which forms the specified diffracting structure in thin-layered light-sensitive coatings . The biggest CGHOE diameter manufactured using this technology reaches up to 300 mm.
Currently, OJSC "NPO GIPO" develops methods and devices for mirrored centered optical systems adjustment of current importance for modern telescope production. One of such methods  developed based on use of CGHOEs with three on-axis computer-generated adjusting holograms manufactured on one joint substrate ensure adjustment control of two-mirror Cassegrain and Ritchey-Chretien telescopes. Optical scheme of the adjustment device is shown on Fig. 11, where CGHOE 6 contains structures of three computer-generated holograms 7, 8 and 9 center of symmetry of which is the point О2. At that, holograms 8 and 9 have ring-shaped apertures. A straight line which passes through points О2 and О3 transversely to CGHOE 6 work surface is a symmetry axis of holograms 7, 8 and 9. It assigns the optical axis of the device when the luminous point (punctual light source) А is brought to this symmetry axis. This is performed when, by the means of linear displacement and angular turns of CGHOE 6 with help of auto collimation hologram 7 automatic collimation image А′7 of the luminous point A is obtained. In this case CGHOE 6 will also be positioned at the specified distance a towards the monochromatic light source A. Computer-generated hologram 8 is a compensating hologram and serves to control the secondary mirror assembly 10 located at the distance d2 from CGHOE 6 and its adjustment in this position transversally to the optical axis and angularly to it. Computer-generated hologram 9 is also a compensating hologram and serves to control the primary mirror assembly 12 at the distance d3 from CGHOE 6 and its adjustment in this position transversally to the optical axis and angularly to it. During the stage of synthesis of holograms 7, 8 and 9 sections a, d1, d2 and d3 are selected; at the same time, a condition is accepted that the sum of d1, d2 and d3 section lengths must be equal to the specified distance d between the peaks О2 and О3 of non-spherical surfaces of the primary and the secondary mirrors of the telescope. When conducting adjustment process of the telescope mirrors, at the final adjustment stage it is necessary to obtain auto collimating images А′7, А′10, А′12 of the luminous point A. This process can be controlled by monitoring 16 images of these points on a display jointly with an image of a control point Ао and their subsequent connection to one another. Setup of 16 images of points or interference bands on the display is made by longitudinal movement of the camera 15.
Study  shows that this method can be used when adjusting Cassegrain telescope with Epps-Shulte focus.
The second method for adjustment control of two-mirror systems of Cassegrain and Ritchey-Chretien telescopes is based on using two on-axis computer-generated holograms applied to a (convex) work surface of a secondary mirror of the telescope on-axis with a side cylindrical surface of the secondary mirror . In this case, a symmetry axis of computer-generated holograms concurs with a symmetry axis of a non-spherical surface of the secondary mirror of the telescope (inaccuracy of non-concurrence of these axes can make up no more than several micrometers). One of such holograms is auto collimating and serves for adjustment of the secondary mirror relatively to a punctual light source, and the second one is a compensating hologram and serves for adjustment of the primary mirror of the telescope relatively to the secondary mirror. Both computer-generated holograms have ring-shaped apertures and can be applied to the edge zone of the secondary mirror outside of its light diameter.
Fig. 12 a shows an optical scheme of a device for implementing of a control method for two-mirror telescope adjustment based on the use of such holograms. Fig. 12 b shows a secondary mirror of the telescope 9 with two ring-shaped computer-generated holograms located at the edge zone of the secondary mirror.
It is worth noting that adjustment holograms 10 and 11 can be also applied to the central zone of the secondary mirror 9 light diameter in case they are manufactured for a wavelength λ which is less than a short-wave border of a spectral range the telescope works in (for example, telescopes for the infrared region of the spectrum). Such positioning of adjustment holograms opens up a possibility to also control adjustment of space-based telescopes during their work on the orbit using this method.
Thus, the current article shows that following methods have been developed, using on-axis CGHOEs:
•a complex of precision methods and means to conduct control of forms and quality of various types of non-spherical (including cylindrical, toric) surfaces of optical components;
•control methods of adjustment of centered lens and mirror systems, including two-mirror telescope systems;
•various applications for solving practical tasks in both optic industry and instrument making, as well as in educational field.
On-axis CGHOEs are basic elements for promising means of adjustment control of two-mirror space-based telescope systems and optical systems of large-dimension objectives of interference instruments for opto-physical research.