We describe a mode sorter for two-dimensional parity of transverse spatial
states of light based on an out-of-plane Sagnac interferometer. Both
Hermite-Gauss (HG) and Laguerre-Gauss (LG) modes can be guided into one of two
output ports according to the two-dimensional parity of the mode in question.
Our interferometer sorts HG_nm input modes depending upon whether they have
even or odd order n+m; it equivalently sorts LG modes depending upon whether
they have an even or odd value of their orbital angular momentum. It functions
efficiently at the single-photon level, and therefore can be used to sort
single-photon states. Due to the inherent phase stability of this type of
interferometer as compared to those of the Mach-Zehnder type, it provides a
promising tool for the manipulation and filtering of higher order transverse
spatial modes for the purposes of quantum information processing. For example,
several similar Sagnacs cascaded together may allow, for the first time, a
stable measurement of the orbital angular momentum of a true single-photon
state. Furthermore, as an alternative to well-known holographic techniques, one
can use the Sagnac in conjunction with a multi-mode fiber as a spatial mode
filter, which can be used to produce spatial-mode entangled Bell states and
heralded single photons in arbitrary first-order (n+m=1) spatial states,
covering the entire Poincare sphere of first-order transverse modes.Comment: 11 pages, 12 figures, 2 appendice