Numerical determination of the substrate’s zero-chirp geometry for the elimination of the period chirp in laser interference lithography

The technique of deliberately bending the substrate during the exposure offers a promising solution to eliminate the period chirp in laser interference lithography. The exact geometry of the substrate to allow for this elimination is given by the solution of an ordinary differential equation (ODE) which has not been solved before. We therefore present a new contemplation of this particular ODE and its solution, the zero-chirp geometry. Considering the planes of constructive interference, we investigated the solution space of the ODE and from this, a more general form of the ODE is developed. Finally, the approach to solve the ODE is described for a specific example, enabling for the first time the determination of the zero-chirp geometry of the substrate to fully eliminate the period chirp in laser interference lithography.