Correction to spatial mode transformation in a modified interferometer

A modified interferometer that introduces position and momentum shifts in mutually orthogonal directions can transform optical modes by raising and lowering radial and azimuthal mode indices. The action of this interferometer can be generalized in the form of a pair of twisting operators, which can be further written in terms of the ladder operators of the 2-dimensional harmonic oscillator. For lower order input modes there is good agreement between theory and experiment, but as the input mode becomes more complex, experimental results start to deviates from first-order theory. This is due to the shifts becoming larger relative to beam structure. We discuss how higher order corrections can be calculated for such cases.