Realization of arbitrary discrete unitary transformations using spatial and internal modes of light

Any lossless transformation on $n_{s}$ spatial and $n_{p}$ internal modes of light can be described by an $n_{s}n_{p}\times n_{s}n_{p}$ unitary matrix, but there is no known procedure to effect an arbitrary $n_{s}n_{p}\times n_{s}n_{p}$ unitary matrix on light in $n_{s}$ spatial and $n_{p}$ internal modes. We devise an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. Our realization uses beamsplitters and operations on internal modes to effect arbitrary linear transformations. The number of beamsplitters required to realize a unitary transformation is reduced as compared to existing realization by a factor $n_{p}^2/2$ at the cost of increasing the number of internal optical elements by a factor of two. Our algorithm thus enables the optical implementation of higher dimensional unitary transformations.