header for SPIE use Propagation modeling for multimode photonics R. P. Ratowsky* a , B. B. Afeyan b , J. S. Kallman a and M. D. Feit a a Lawrence Livermore National Laboratory, Livermore, CA 94550 b Polymath Research, Inc., Pleasanton, CA 94566 ABSTRACT While ray optics is a valid model for many multimode optical systems, it cannot capture diffraction and interference phenomena which may be required, for example, in calculating speckle propagation in multimode fiber. Wave optics includes the effect of interference and diffraction, but it is usually much more costly computationally. We show how to bridge the gap between ray and wave optics by using a position-angle ray phase space representation of the electric field. By transporting a suitable quadratic functional of the field (the Wigner distribution) along rays, diffraction and interference are taken into account for propagation through an arbitrary transverse index profile. We show how this method allows us to propagate highly multimode fields without resorting to detailed mode calculations. We also illustrate the method by calculating the mode selective loss for a multimode graded index waveguide. 1. INTRODUCTION Interest in multimode photonics systems is widespread and growing rapidly, motivated in part by ease of alignment and large numerical apertures for light collection. Examples include computer backplane communications, local area networks, and multimode interferometry. The simulation of multimode systems presents additional challenges beyond those of the single-mode case. First-principles propagation calculations can be costly except over very short distances because of the fine zoning required to accommodate a large number (thousands) of modes, and the lack of knowledge of mode-coupling perturbations. Furthermore, waveguide dispersion (absent in the single-mode case) must be assessed. Ray tracing may be able to give accurate results for energy transport but ignores phase: an accurate calculation of coupling efficiencies (e.g. VCSEL to fiber) and mode-selective loss requires capturing the details of the multimode speckle pattern. Radiation transport approaches may also be used, but this requires calculating a large number of modes and eigenvalues through WKB or some other method. We demonstrate here the accurate propagation of speckled optical fields, without resorting to detailed mode calculations, by propagating rays in their position-angle phase space. By transporting a suitable quadratic functional of the electric field (the Wigner distribution), diffraction and interference are taken into account for propagation through an arbitrary transverse index profile. We can therefore exploit the efficiency of ray propagation over full wave optics for highly multimode systems, while retaining needed wave-optical physics. The method can be used to advantage in characterizing multimode photonics propagation and phenomena such as mode-selective loss. This paper is organized as follows. First, we will convey the principles of the phase space method and give some simple examples. Then, we will describe the software tool we developed to study the propagation of fields through the Wigner method. This tool allows us to study the important issue of the appropriate sampling of rays in the phase space to achieve desired accuracy. Further examples of light propagation in systems with a variety of axial and transverse refractive index distributions will be given. We then show how the method can be applied to calculate modal noise in a multimode graded-index waveguiding system. Finally, we will describe some of the limitations of our current scheme and the path we envision for further development and applications. 2. BASIC FORMULATION In this section we will develop some of the properties of the phase space formalism we will need, in an elementary way. We usually think of an electric field as a vector quantity which varies in space and time according to equations. We can also * Correspondence: rpr@llnl.gov; 925.423.3907 voice; 925.423.5080 fax