Free space and waveguide Talbot effect: phase relations and planar light circuit applications

Abstract

Optical fields that are periodic in the transverse plane self-image periodically as they propagate along the optical axis: a phenomenon known as the Talbot effect. A transfer matrix may be defined that relates the amplitude and phase of point sources placed on a particular grid at the input to their respective multiple images at an image plane. The free-space Talbot effect may be mapped to the waveguide Talbot effect. Applying this mapping to the transfer matrix enables the prediction of the phase and amplitude relations between the ports of a Multimode Interference (MMI) coupler– a planar waveguide device. The transfer matrix approach has not previously been applied to the free-space case and its mapping to the waveguide case provides greater clarity and physical insight into the phase relationships than previous treatments. The paper first introduces the underlying physics of the Talbot effect in free space with emphasis on the positions along the optical axis at which images occur; their multiplicity; and their relative phase relations determined by the Gauss Quadratic Sum of number theory. The analysis is then adapted to predict the phase relationships between the ports of an MMI. These phase relationships are critical to planar light circuit (PLC) applications such as 90° optical hybrids for coherent optical receiver front-ends, external optical I-Q modulators for coherent optical transmitters; and optical phased array switches. These applications are illustrated by results obtained from devices that have been fabricated and tested by the PTLab in Si micro-photonic integration platforms.