JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 5, MAY 1997 815 Analysis of Integrated Optical Waveguide Mirrors R´ egis Orobtchouk, Suzanne Laval, Daniel Pascal, and Alain Koster Abstract— A new method is presented to analyze reﬂection losses of integrated mirrors, taking into account the exact guided mode proﬁle and assuming that this proﬁle remains unchanged up to the reﬂecting plane. The fraction of the reﬂected light coupled to one of the guided modes of the output waveguide is calculated, taking into account the mirror reﬂection coefﬁcient. The inﬂuence of both translation and tilt of the reﬂecting plane is investigated. The method applies for every guided mode and any reﬂection angles. Numerical calculations are derived for a 90 optical corner mirror. I. INTRODUCTION C HANGING the direction of optical guided beams over a short distance is a key point to increase device density in the implementation of photonic integrated circuits. Curved bending waveguides have been proposed [1] but they need a large curvature radius to keep the radiation loss acceptable and are then very space consuming. Small radii of curvature are possible using strongly guiding structures [2], [3] which in turn are less adapted to efﬁcient coupling with single mode optical ﬁbers. One way to get abrupt directional change in the propagation direction is to make use of a waveguide mirror [4], [5]. Improvements in fabrication processes, which allow loss reduction [6], increase the interest for such devices. Several theoretical analysis were reported and used to estimate the inﬂuence of fabrication tolerances [4], [7], [8] for optical corners. The main difﬁculties consist in properly taking into account the guided characteristics of the incident beam near the mirror surface and the mirror nonideality. The beam propagation method [8], which is widely used to simulate electromagnetic ﬁeld propagation in integrated optical circuits, is not really suitable when waveguides are tilted with respect to the propagation axis [9]. Plane wave expansion has been derived with various ap- proximations [5], [7], [10]. It is generally assumed that near the mirror the incident beam propagates in a homogeneous medium whose refractive index is equal to the equivalent index of the waveguide core material, as deﬁned by the effective index method to describe two-dimensional (2-D) conﬁnement. This approximation may be valid for weak ﬁeld conﬁnement and small beam divergence (paraxial approximation). How- ever, even in these cases, this approximate method does not give satisfaction to calculate for example the facet reﬂectivity of semiconductor laser diodes [11] and an equivalent refractive index has been proposed for the homogeneous medium [12]. Furthermore, most of the authors analyze the inﬂuence of a Manuscript received April 22, 1996; revised November 18, 1996. This work was supported by CNET-France Telecom. The authors are with the Institut d’ ´ Electronique Fondamentale, CNRS URA 22, Universit´ e Paris Sud, 91405 Orsay Cedex, France. Publisher Item Identiﬁer S 0733-8724(97)03535-4. small mirror tilt, but do not consider a possible translation of the reﬂecting facet from the ideal symmetrical position at the intersection of the waveguide axes. Translation and tilt have never been considered simultaneously. The method described in this paper is based on plane-wave expansion. It uses an original description of the incident guided beam which frees one of the paraxial approximation and of the homogeneous medium assumption. It assumes that the mode shape is not altered by the mirror until light is reﬂected. It can be applied whatever the angle between the two waveguides is, for strongly guiding as well as for weakly guiding structures, and is not restricted to the fundamental guided mode. The fraction of light coupled to each mode of the output waveguide can be calculated. The inﬂuence on the loss level of both mirror translation and tilt is considered. The effect of mirror surface roughness is also investigated. II. METHOD OF ANALYSIS A cross section of the structure under consideration is represented in Fig. 1. A reﬂecting plane intersects the input and output waveguides. Its reference position (broken line in the ﬁgure) is symetrical with respect to the waveguide axes, with an angle between the mirror normal and each axis, and with the intersection point of the waveguide axes located in the mirror plane. The reﬂecting plane can be translated by a distance and/or tilted by an angle with respect to this reference position. It is convenient to use three coordinate axis systems (Fig. 1): the incident guided wave propagates along the direction in the system, the coordinates are linked to the mirror and the system is used for the output waveguide. The origin of the axes is labeled when the mirror is placed in its ideal symetrical position. In the general case, the origin is chosen on the bisecting line of the waveguide axes and is deﬁned by Using the effective index method approximation [13], [14], the 2-D waveguides are reduced to slab ones. Quasi-TE-modes of the 2-D waveguides correspond to a -polarized component of the magnetic ﬁeld, i.e., a TM tranverse conﬁnement, and for quasi-TM modes (TE transverse conﬁnement) the electric ﬁeld is along the axis. The equivalent refractive index is in the guide core and in the surrounding material. Symmetrical waveguides are considered as the index is the same on each side of the guiding core, but the method is easily generalized to nonsymmetrical waveguides [15]. Leaving out the time dependence the guided incident electromagnetic ﬁeld in the input waveguide ( for quasi-TM modes or 0733–8724/97$10.00  1997 IEEE