Integrated holographic vector matrix multipliers

We show that integrated optical volume holographic devices may implement arbitrary linear transformations with order unity efficiency. At the diffraction limit, the device area needed to implement an N × N transformation is proportional to N²λ², but in materials with a maximum index modulation of Δn_max, the area scales as N³²λ²Δn_max. This scaling represents a factor of a N improvement over two-port device networks, such as Mach-Zehnder arrays. A holographic system may thus implement 1000 × 1000 transformations in 30 × less area and 10, 000 × 10, 000 transformations in 100 × less area. We explore the practical implications of these scaling laws in the design of electro-optically controlled holographic vector matrix multipliers.