Programmable holographic technique for implementing unitary and nonunitary transformations

Beyond the possibilities of linear transformations in polarization space, whose dimensionality is constrained by limited orthogonal states, we propose a technique for implementing both unitary and nonunitary transformations with higher dimensionality. Any high-dimensional matrix can be decomposed into a product of two processes realizable by utilizing spatial phase modulation and free-space propagation, in a simple, fixed, and scalable setup. Given that perfect power transmission for an arbitrary matrix may not be possible, the method is optimized to reach the theoretical best. Projected applications of the method described here include a means of restricting the infinite-dimensional Hilbert space to a finite-dimensional basis for information processing purposes, simultaneous multichannel optical routing, and a method of optical orbital angular momentum sorting and generation.

Figure 1

The typical amplitude patterns of (a) angle and (b) OAM states, and (c) quasiangle and (d) quasi-OAM states with $N=6$. The angle state is approximately illustrated by the superposition of $\sim 200$ OAM states.

Figure 2

Implementation principle of matrix transformation by two transmissive SLMs (SLM1/SLM2), a $2f$ focusing lens system (L1/L2), and a pinhole, of the simplest case of $N=2$ for clarity.

Figure 3

Proposed schematic setup for implementing unitary and nonunitary transformations of $N=5$.

Figure 4

(a) Principle of the checkerboard approach and (b) the paired adjacent pixels as superpixels on phase-only SLM.

Figure 5

Shift matrix and clock matrix transformations with (a) quasiangle state and (b) quasi-OAM state.

Figure 6

(a) Permutation matrix and (b) tridiagonal Toeplitz matrix transformations.

Figure 7

Topological charge measurement of Laguerre-Gaussian beams with DFT matrix transformations.

Figure 8

Generation of quasi-OAM state from quasiangle state with inverse DFT matrix transformation.