Generalized optical interferometry for modal analysis in arbitrary degrees of freedom

Abstract

We generalize the traditional concept of temporal optical interferometry to any degree of freedom of a coherent optical field. By identifying the structure of a unitary optical transformation that we designate the generalized phase operator, we enable optical interferometry to be carried out in any modal basis describing a degree of freedom. The structure of the generalized phase operator is that of a fractional optical transform, thus establishing the connection between fractional transforms, optical interferometry, and modal analysis.

Basis-neutral Hilbert-space analyzers

“…For a discrete modal basis indexed by n (Equation 1), is periodic in α with period 2 π . Furthermore, Λ can be generalized to two transverse coordinates and is applicable to a continuous basis3334.…”

“…We introduce the concept of a generalized delay (GD): an optical transformation characterized by a continuous, real order-parameter that can be tuned to produce – once placed in one arm of an interferometer – an interferogram that reveals the modal weights in a prescribed functional basis via harmonic analysis. We find that GDs correspond to optical implementations of fractional transforms in the case of discrete modal bases3334. For example, it can be shown34 that the GD associated with Hermite Gaussian (HG) modes is the fractional Fourier transform3536, whereas that associated with radial LG modes37 is the fractional Hankel transform3839.…”

“…We find that GDs correspond to optical implementations of fractional transforms in the case of discrete modal bases3334. For example, it can be shown34 that the GD associated with Hermite Gaussian (HG) modes is the fractional Fourier transform3536, whereas that associated with radial LG modes37 is the fractional Hankel transform3839. Sweeping the order of a fractional transform corresponds to varying a temporal delay in traditional interferometry – each in its own Hilbert space.…”

Two-point optical coherency matrix tomography

“…The principle of this method may be extended to other DOFs (such as optical orbital angular momentum or other spatial modes,) by using appropriate projective measurements [15,16]. This result hinges on the fact that the vector space describing the properties of an electromagnetic field having multiple DOFs [5,17] is the direct product of the vector spaces corresponding to each DOF.…”

Phase-unlocked Hong-Ou-Mandel interferometry

“…Finally, the HOM-I and PhuL HOM-I may be extended to other degrees of freedom, such as orbital angular momentum, by replacing the delays with so-called general phase operators [22], thereby enabling the analysis of 2P states in an arbitrary basis [23].…”