Geometrical optics can be defined as a local plane wave approximation of optical wave theory. In diffractive optics it can be used for two purposes. In the first place the concept of thin phase or amplitude objects can be elucidated by considering the boundaries between geometrical optics and wave theory. In the second place the transformation of waves by diffractive optical elements can be descreibed by geometrical optics. We will treat in the present paper both problems. For the description of propagation and imaging we use the method of characteristic functions of Hamilton. We will show that it is possible to describe diffractive elements by a characteristic function or a sum of characteristic functions. We will apply Hamilton''s method to the calculation of image quality and tolerances of diffractive elements.

Holographic optical lens elements (HOE) normally suffer from the aberrations and deviations from the Bragg condition caused by the inevitable wavelength mismatch between recording and reconstruction. A possibility to calculate the phase functions, which are necessary to precompensate in the recording step both the aberrations and the deviations from the Bragg condition, is presented and numerical examples are shown. The phase functions can be generated with the help of computer generated holograms (CGH).

The wave aberrations determine the quality of the focal spot and more general the imaging quality of the lens under test. Here we propose the measurement of the wave aberrations with the help of a Twyman-Green interferometer adapted to the special requirements for testing holographic optical lens elements. The evaluation of the interferograms is done with the phase-shifting technique. The resulting wave aberrations are expanded as Zernike polynomials. In addition to this evaluation the point spread function and the modulation transfer function are calculated from the wave aberrations. The setup the evaluation method and some exemplary results of a tested holographic optical element are presented.

Computer generated diffractive optical elements, known as "focusators," provide for very complicated focal intensity distributions, depending on the number of pixels and quantization levels. The manufacturing of such elements is very expensive and must be preceded by computer simulation of their performance. Computer experiment with laser beam focusators into segment is described in this paper. The estimations of power efficiency and focal line quality for various parameters of the phase digitization are obtained. The different methods of focusator synthesis are analyzed.

The use of scalar diffraction theory is the basis of a fundamental part of diffractive optics. We prefer to call this part digital holography. The consideration of different parts of diffractive optics on the basis of the diffraction theory seems to be a useful criterion to discuss fundamental characteristics of diffractive elements. In digital holography a coding theory has been developed. The theory itself and its consequences are briefly described.

We calculate quantized digital holograms with three iterative techniques, namely simulated annealing, direct binary search , and Fourier transform algorithm. The behavior of the algorithms depends significantly on the initial distributions being used. With the help of coding theory in digital holography advantageous initial distributions can be found, that improve the capabilities of all three techniques. This paper is based on reference [1].

Iterative Fourr Transform Algorithms provide an excellent tool for calculation of analog and quantized synthetic holograms. In principle the distributions in the object and the Fourier domain are modified to meet constraints given in both domains. In many practal situations these transformations of data belong to the class of the so called point tnuisformaticns which generally do not preserve L2 norm of data their total power or energy. It will be shown the usage of global transformations supplemented by tl ccnrvation constraint offers sorr advantages to point transformations with respect to the convergence of the algorithm. Resuitsof several numerical experiments will be given.

The calculation of high quality digital holograms with the error diffusion concept is presented. The cont''nts of this paper is based on reference

We present a new strategy for the calculation of diffractive elements with high efficiencies and high signalto- noise-ratios. The procedure is based on coding theory in digital holography. We discuss three different modulation schemes for diffractive structures and present first experimental results. I.

Computer generated holograms for display have been developed in the last decade but many numerical difficulties arise when the Fresnel and Fraunhofer hypotheses do not hold. For display applications, interference between different points of the object can be considered as intermodulation noise. Thus, we consider intermodulation-noise-free intensity holograms. Three different numerical methods have been used to generate holograms of warped curves: point-to-point, a specific numerical integration, and stationary phase approach. The most flexible one is the point-to-point method that takes advantage of the linearity of intermodulation-noise-free holograms, but very large computation times are attained. Integrated methods are more economic and different elements of curves can be easily added to built up figures, but specific algorithms have to be devised for each kind of curve. Some examples of computer generated intermodulation-noise-free holograms of objects composed by linear segments are presented as well as reconstructions of them.

As a result of the continuously improving resolution of the VLSI-based micro-optic fabrication technology, diffractive optical elements can now be manufactured that no longer obey the usual physical optics boundary conditions and the paraxial Fresnel and Fraunhofer diffraction integrals. Rigorous electromagnetic diffraction theory of gratings is applied here to predict the limits of the approximate theory, and to analyze and design diffractive microelements that can be treated only within the framework of the rigorous theory.

A hybrid hologram is a volume hologram recorded interferometrically, using the output wavefront of a computer-generated hologram as an object wave. It thus has the advantage of high diffraction efficiency of a volume hologram and the flexibility of a computer-generated hologram. We apply the thin grating decomposition method for the numerical analysis of the properties of hybrid holograms. The effects of several parameters in recording and reconstruction on the quality of the regenerated image are investigated using hybrid kinoform beam array generators as an example.

We have designed a range of computer-generated guided-wave kinoform optical elements using the thin-grating decomposition method and nonlinear optimization. Kinoform (Fresnel) lenses with improved local and overall diffraction efficiency have been constructed. High- efficiency, low-noise guided-wave array generators with fan-out up to 16 have been optimized.

We report on direct-writing EBL manufactured, proximity compensated blazed transmission gratings. The proximity compensation is made using a non-linear iterative process in the spatial domain. The diffraction efficiency for a compensated 8 micron period grating was 84%, almost twice that of an uncompensated grating.

Computer generated holograms (CGH) consist of patterns which have been calculated numerically starting from the desired function of the optical component and the known physical laws of diffraction. The real fabrication of such holograms poses two different tasks. First the computed pattern has to be converted into one mask or a set of masks. Thereby submicron resolution is required for CGHs that are to be used for visible light. This means that one has to use microlithographic techniques such as electron-beam or laser lithography. The second task is to achieve the necessary groove depth and profile. Wet etching techniques have been quite common for this purpose but they have two disadvantages: it is difficult to obtain a precise etch stop particularly with very fine structures due to capillary effects and mostly one is constrained to certain crystallographic orientations. This makes it hardly possible to achieve directional anisotropic etching in glass. Dry etching techniques overcome both of these problems. The foundry for micro-optical elements in Erlangen makes use of laser lithography and reactive-ion etching. Both technologies are discussed in some detail. The scope of computer generated holograms ranges from the well known Dammann-gratings, gratings which produce arrays of light spots of equal intensity, to very complex two- dimensional structures for example for the correction of aberrations or for three-dimensional displays. In his paper we also describe some results with the production of microlenses that can be realized using our technologies.

New technologies for fabricating optical micro- and nanostructures enable the realization of planar diffractive optical elements (DOEs). Almost any structure shape, including non- rotational aspherics, can be manufactured, which provides all degrees of freedom for the design. This paper summarizes the design nd fabrication of DOEs for beam shaping and imaging. Selected applications include holographic laser scanners, 2-D fan-out elements by continuous surface relief-elements, and beam shaping of high-power laser diode arrays.

In this paper we discuss the design, fabrication, and performance of multi-level grating array illuminators. A complete fabrication error analysis is carried out for one dimensional designs with sixteen equally spaced relief levels. Array illuminators with eight/sixteen relief depth levels are fabricated in fused silica using reactive ion etching and photolithography with three/four electron bema binary amplitude masks. Fanouts as large as 32 X 16 and efficiencies > 90% have been achieved, which to our knowledge are the largest and most efficient realized to date using multi-level relief structures.

In this contribution a few simple examples are presented in which diffractive optical elements are applied within optoelectronic demonstration systems for interconnection purposes and for optical sensors. Both volume holograms recorded as interferograms with large deflection angles and high diffraction efficiencies are applied in these examples, as well as synthetic holograms fabricated as surface relief gratings by microlithography which offers a maximum freedom in design.

Diffractive lenses can be obtained by recording the interference pattern originated by interference of two different spherical waves. The imaging quality of such optical elements usually is described in terms of III-order aberrations. The aberration coefficients depend on the radii of curvature of waves used for the lens recording and location of its input pupil. The imaging quality also depends on such factors as image contrast and background illumination level (due to scattered light). Those factors do not result from aberrations, but are dependent on the type of recording process. Namely, light diffraction occurs on the system of fringes of profile depending on the transmittance-versus-exposure characteristics of the recording material. Several types of lenses including kinoform, linear, nonlinear, and binary amplitude, as well as linear and binary phase diffractive lenses of the same III-order aberrations, are investigated. Point spread function and incoherent transfer function numerically calculated are compared. The other factor influencing the imaging quality is modulation transfer function of the recording material. The different local spatial frequency of the diffractive lens microstructure corresponds to the different local diffraction efficiency. This effect is similar to apodising and also can change the imaging characteristics. Four typical material modulation transfer functions are considered: linear, parabolic, hyperbolic, and band-pass -- typical for photothermoplastics. The resulting point spread functions and incoherent transfer functions are calculated for diffractive lenses with aberrations: one having uncompensated astigmatism, the other with considerable coma.

Binary phase gratings (Dammann gratings) are introduced as a new standard optical component within laser material processing applications and are shown to offer considerable advantages.

In this paper we report on the design of diffractive optical elements (DOEs) for high power laser radiation. We modified the Fourier transform algorithm which allows hologram calculation also for reflective type Kinoforms considering the tilted arrangement. To achieve light weighted DOEs which are resistend against intense laser radiation reflective DOEs in silicon were fabricated. The elements have been investigated with respect to the diffraction efficiency and the absorption values.