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Optics Letters

Generalized optical interferometry for modal analysis in arbitrary degrees of freedom

Ayman F. Abouraddy, Timothy M. Yarnall, Bahaa E. A. Saleh

Published Jul 10, 2012• 25 citations
Publisher

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.

DOI
10.1364/ol.37.002889

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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.
Via the route of developing Dirac's symbolic method and following Dirac's assertion: "⋯ for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory", we find the generalized Fresnel operator (GFO) corresponding to the generalized Fresnel transform (GFT) in classical optics. We derive GFO's normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in quantum optics. With the aid of entangled state representation the entangled Fresnel transform is proposed; new eigenfunctions of the complex fractional Fourier transform and fractional Hankel transform are obtained; the two-variable Hermite eigenmodes of light propagation are used in studying the Talbot effect in quadratic-index media; the complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Moreover, quantum optical version of classical z-transforms is obtained on the basis of the eigenvector of creation operator. Throughout our discussions, the coherent state, squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used.
We extend the theory of matter-wave interferometry of pointlike particles to nonspherical objects by taking the orientational degrees of freedom into account. In particular, we derive the grating transformation operator, which maps the impinging state onto the outgoing state, for a general, orientation-dependent interaction potential between the grating and the molecule. The grating operator is then worked out for the diffraction of symmetric top molecules from standing light waves, and the resulting interference pattern is calculated in the near field. This allows us to identify a signature of the orientational degrees of freedom in near-field matter-wave experiments.
Entanglement is a unique feature of quantum mechanics. In coupled systems of light and matter, entanglement manifests itself in the linear superposition of multipartite quantum states (e.g., parametrized by multiple spatial, spectral, or temporal degrees of freedom of a light field). In bipartite systems, the Schmidt decomposition provides a modal decomposition of the entanglement structure over independent, separable states. Although ubiquitous as a mathematical tool to describe and measure entanglement, there exists no general efficient experimental method to decompose a bipartite quantum state onto its Schmidt modes. Here, we propose a method that relies on bipartite self-configuring optics that automatically “learns” the Schmidt decomposition of an arbitrary pure quantum state. Our method is designed for entanglement distributed over spatial or spectral degrees of freedom of a light field and can reconstruct the Schmidt modes and values by variational optimization of the network’s output powers or coincidences. We illustrate our method with numerical examples of spectral entanglement analysis for biphotons generated via spontaneous parametric down conversion and provide experimental guidelines for its realization and generalization to other degrees of freedom of light, including the influence of losses and impurities. Our method provides a versatile and scalable way of analyzing entanglement in bipartite integrated quantum photonic systems.
Optical coherence is a well-established branch of physical optics in which the statistical properties of fluctuating optical fields are described in terms of correlation functions over continuous spatial and temporal degrees of freedom (DoFs). Nevertheless, in any practical setting, only discrete DoFs are ever accessible. In these scenarios, the modes are fixed, stable, and deterministic, and partial coherence arises from random relative complex amplitudes, a configuration we refer to as 'structured coherence'. Advances in structured coherence have recently unveiled new conceptual ground in optical communications and information processing in which partial coherence may be preferable to full coherence, which we call a 'coherence advantage'. We formulate structured coherence in terms of coherence matrices to investigate these recent theoretical and experimental breakthroughs. We first review optical fields characterized by a binary DoF via 2x2 Hermitian coherence matrices and introduce key concepts that take on new significance for larger-dimensional modal sets. Next, we examine the structured coherence of two binary DoFs, which can be described by 4x4 coherence matrices. We introduce the concept of coherence rank, entropy swapping, and optical cross-purity. In the perspective outlined here, coherence is viewed as a 'resource', which can be exchanged between DoFs, concentrated into a DoF or into particular modes, or spread over the DoFs. We then examine larger-dimensional modal sets, which allow for more versatile applications in optical information processing. The formulation presented here lends itself particularly to the manipulation of partial optical coherence in integrated photonic platforms, thereby opening myriad avenues for novel fundamental investigations of structured coherence and exploiting the coherence advantage in optical communications and information processing.