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Realization of arbitrary discrete unitary transformations using spatial and internal modes of light
Ish Dhand, Sandeep K. Goyal
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Any lossless transformation on nₛ spatial and nₚ internal modes of light can be described by an nₛnₚ×nₛnₚ unitary matrix, but there is no known procedure to effect an arbitrary nₛnₚ×nₛnₚ unitary matrix on light in nₛ spatial and nₚ 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 nₚ²/2 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.
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