Transverse-mode beam splitter of a light beam and its application to quantum cryptography

We have theoretically and experimentally studied how a Mach-Zehnder interferometer with an additional mirror transforms a light beam composed of the second lowest transverse modes, ${\mathrm{HG}}_{10},$ ${\mathrm{HG}}_{01},$ ${\mathrm{LG}}_{01},$ and ${\mathrm{LG}}_{0-1}$ (HG denotes Hermite-Gaussian mode; LG denotes Laguerre-Gaussian mode). In certain conditions, the interferometer divides the incident beam into the ${\mathrm{HG}}_{10}$ and ${\mathrm{HG}}_{01}$ components as a transverse-mode beam splitter. We propose a practical device involving the two interferometers for quantum cryptography, in which a photon carries two bits corresponding to the polarization and the transverse mode.