Figure 1

The interferometric implementation of the X gate together with its integer powers. (a) The OAM exchanger \(\mathrm{EX}_{k}\) of order k is built from two holograms and a Leach interferometer [22] with one Dove prism rotated through \(\pi /(2 k)\). Optical elements: holo—hologram, Dove—Dove prism, BS—50:50 beam splitter. (b) The inverse of the OAM exchanger, \(\mathrm{EX}_{k}^{-1}\), has almost the identical structure to that of \(\mathrm{EX}_{k}\) only the Dove prism is rotated through \(-\pi /(2 k)\). For convenience, we use two slightly different symbols to denote the inverse of the OAM exchanger, as shown in the figure. (c) The X gate for \(d = 8\). (d) The X gate from (c) is constructed from two OAM sorters, marked by shaded rectangles in the figure, from which redundant exchangers are removed. These exchangers can be grouped into blocks of increasing size, which are enclosed in dashed-line rectangles. The unused paths as well as redundant exchangers are drawn in faded color. The remaining exchangers can be reordered in order to get rid of the path permutations. (e) The same principles apply when constructing the integer powers \({X}^{k}\) of the X gate. When the exponent k is a power of two, i.e., \(k = 2^{m}\), the path permutation has a repetitive structure and the whole setup is effectively split into k identical subsetups. (f) For a general exponent k the path permutation has a more complicated structure. (g) The number of exchangers that have to be retained in the final setup increases with the exponent k until it attains the form \(k = d/2\), in which case no exchangers can be removed