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The geometry of refraction and interference of two plane laser beams into a flat recording package is sufficiently simple that analytic expressions can be derived for the beam and sample angles to write a given transmission or reflection hologram. The accessible range of grating vectors is constrained only by the coherent transfer function determined from the cut-off angle of writing ray vectors. To reach a larger portion of the grating space, as is typically required for materials testing and applications such as augmented reality waveguide couplers, one must add prisms to one or both sides of the sample. The constraints on possible grating vectors in this case are a complex interplay of the cut-off angle, the clear aperture of the prism faces, hologram position in the sample, and the limits of the motion system that positions the recording medium and writing beams. Further, the number of possible writing geometries jumps from two in the planar case (that is, reflection and transmission) to the square of the number of entry facets or 16 in the case of triangular prisms on both sides of the sample. Determining the best prism, writing beam, and stage configuration to write a required set of gratings at a multiplicity of locations in the sample thus becomes analytically intractable.
We present a graphical method of solving this inverse problem. The foundation is a highly parallel, non-sequential ray tracer that solves for all possible holographic gratings given a specific sample geometry consisting of a film, substrates, and triangular prisms on one or both sides. The tool graphically presents the allowable grating vectors as a function of writing beams entering surfaces in all possible combinations, hologram position in the sample, and motion control such as writing beam and sample rotation and translation. Constraint regions due to the cut-off angle, surface clear apertures, and stage motion are illustrated to guide design modifications that would reach different regions of grating space. Finally, we show how this supports writing into curved films.
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