Analysis of Second-Order Gratings

We report the results of a second-order grating analysis. The gratings are used as distributed Bragg reflectors in surface-emitting lasers, which are currently being fabricated in several laboratories. The gratings provide reflection, output coupling, and power transmission to other gain segments for purposes of injection locking. The analysis determines these quantities for arbitrary-shaped grating teeth and includes the presence of a substrate reflector to reduce the radiated power in that direction. The reflector is shown to be effective, but only if it can be precisely positioned. Examples illustrating variations in dimensions, tooth shapes and heights, waveguide loss, and detuning are included. Second-order, square gratings, whose conventional coupling coefficient is identically zero, exhibit substantial reflectivity which is primarily produced by radiation reaction. The analysis also forms the basis for calculating the far-field grating-coupled radiation patterns.