Volume 59, number 5,6 OPTICS COMMUNICATIONS 1 October 1986 zyxwvuts A COUPLED WAVE ANALYSIS FOR ON-AXIS HOLOGRAPHIC LENSES IN GENERALIZED COORDINATES E. GUIBELALDE Cdtedra de Rsrca Mkdica, Facultad de Medicina, Universidad Complutense, 28040 Madrid, Spam and M.L. CALVO ’ Departamento de Optica, Facultad de Ciencias Fkicas, Unioersidad Complutense, 28040 Madrid, Spain Received 17 March 1986 A coupled wave analysis is presented to study the diffraction of light by on-axis holographic lenses. The mathematical procedure is carried through by using generalized coordinates. Analytical and numerical results are evaluated, allowing a physical interpretation in terms of waves geometry. 1. Introduction Holographically formed lenses are of considerable importance in optic communication due to their po- tential use as couplers [l-4] , beam expanders, and holographic optical devices for integrated optics. In the context of the possible mathematical procedures, 3D-rigorous theories [5] lead to partial differential equations which may be solved with a great mathemat- ical and numerical effort [6] . Local 1 -D theory (which is indeed much more simple) may be used if the thick- ness of the lens is assumed to be small compared to its focal length [6] which is true for most practical lenses, however, couplers or end fibers lenses might be with focal lengths of the order of its thickness. In this communication a coupled wave equation is proposed in generalized coordinates to be used for on-axis holo- graphic lenses in the whole range of focal lengths and thicknesses. Although the equations can be extended to higher diffraction orders, in the present paper, we only consider two coupled waves. The solution is given analytically up to second order of the modulation ’ Address for the Academic year 1985-86, School of Optome- try, University of California, Berkeley, CA 94720, USA. 0 0304018/86/SO3.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) Fig. 1, Geometry of the recording lens hologram. The model has been developed for an infinite medium (d = m). constant and numerical estimates are presented. Fig. 1 shows the recording geometry in the plane of interest. The state of polarization is supposed to be perpendicular to this plane for all recording and recon- structing waves. 2. Coupled wave analysis By interfering the plane reference wave and the spherical wave of fig. 1, a dielectric permittivity varia- tion of the following form can be assumed in the re- 331