Volume 81, number 6 OPTICS COMMUNICATIONS 15 March 1991 Rigorous diffraction analysis of Dammann gratings Antti Vasara, Eero Noponen Department o f Technical Physics, Helsinld University o f Technology, 02150 Espoo, Finland Jari Turunen, J. Michael Miller and Mohammad R. Taghizadeh Department of Physics, Heriot-Watt University, Riccarton, Edinburgh EH I 4 4AS, UK Received 28 September 1990 Rigorous diffraction theory is applied for the first time to the analysis of periodic, binary computer-generatedholographic optical fan-out elements, often called Dammann gratings. The effects of the length of the grating period on the reconstruction noise and the diffraction efficiencyare analyzed and discussed. Novel schemes are proposed for producing compact and highly efficient space-invariantoptical interconnectsthat operate in the resonancedomain. 1. Introduction The rapidly expanding effort towards the realiza- tion of highly parallel digital optical processors has prompted a need to develop efficient optical devices that generate one- or two-dimensional arrays of equal- intensity light spots. Certain binary Fourier domain computer-generated holograms, the so-called Dam- mann gratings [ 1,2 ], have become standard devices in the field of space-invariant array generation [ 3,4 ]. Gratings with large fan-out (up to 201 ×201 ) have been demonstrated [5 ],, which show good recon- struction fidelity (better than + 10%). The design and analysis of Dammann grating s has so far been based on scalar diffraction theory and Kirchhoff approximations. Within this formalism, the amplitudes of the diffraction orders are given by the Fourier coefficients of the grating transmission function [ 6 ]; they are thus fully characterized by the phase delay and a set of phase transition points within the grating period [ 1-5 ]. The desired diffraction pattern with N equally bright central orders can therefore be obtained by suitably adjusting these free parameters with the aid of nonlinear optimization techniques such as simulated annealing [ 3 ]. The usual measures of quality characterizing a given grating profile are the diffraction efficiency, i.e. the amount of transmitted power that is diffracted in the desired N orders, and the reconstruction error (or array uniformity), defined as the maximum de- viation of a signal beam power from the average power. However, a third factor, namely the mini- mum feature size AXmin/d within the grating period (length d) is also of importance. It is intuitively clear that this parameter must be, at least roughly, in- versely proportional to N. This tendency has indeed been verified by optimizing a large number of Dam- mann gratings with different values of N (some of the results are collected in table 1 ). The Dammann grating periods presently required a range from about 100 ~tm to a few millimeters. The low limit is ap- propriate in miniature systems based on fiat [ 7 ] or stacked [ 8 ] optics, while the millimeter-range pe- Table 1 Fourier optics predictions for the characteristics of the Dam- mann gratings considered in figs. 2-7: diffraction efficiencyPE, array uniformity AR, average feature size Ax/T, and minimum feature sizeAXmiJ T are given. N ~R% P~% Ax~n/T ~Lx/T 7 0.43 78.7 0.10 0.25 15 0.71 81.8 0.06 0.12 33 0.12 80.8 0.02 0.06 8 0.86 71.5 0.05 0.12 16 0.65 80.6 0.02 0.06 32 0.26 81.8 0.01 0.03 0030-4018/91/$03,50 © 199i - Elsevier Science Publishers B.V. ( North-Holland) 337