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arXiv:1912.10109 · Journal of the Optical Society of America B

Theory of four-wave mixing of cylindrical vector beams in optical fibers

E. Scott Goudreau, Connor Kupchak, Benjamin Sussman, Robert W. Boyd, Jeff S. Lundeen

Published Apr 2, 2020• 11 citations
arXivHTMLPDF

Abstract

Cylindrical vector (CV) beams are a set of transverse spatial modes that exhibit a cylindrically symmetric intensity profile and a variable polarization about the beam axis. They are composed of a non-separable superposition of orbital and spin angular momenta. Critically, CV beams are also the eigenmodes of optical fiber and, as such, are of widespread practical importance in photonics and have the potential to increase communications bandwidth through spatial multiplexing. Here, we derive the coupled amplitude equations that describe the four-wave mixing (FWM) of CV beams in optical fibers. These equations allow us to determine the selection rules that govern the interconversion of CV modes in FWM processes. With these selection rules, we show that FWM conserves the total angular momentum, the sum of orbital and spin angular momenta, in the conversion of two input photons to two output photons. When applied to spontaneous FWM, the selection rules show that photon pairs can be generated in CV modes directly and can be entangled in those modes. Such quantum states of light in CV modes could benefit technologies such as quantum key distribution with satellites.

DOI
10.1364/josab.386622

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Cylindrical vector (CV) beams are a set of transverse spatial modes that exhibit a cylindrically symmetric intensity profile and a variable polarization about the beam axis. They are composed of a non-separable superposition of orbital and spin angular momenta. Critically, CV beams are also the eigenmodes of optical fiber and, as such, are of widespread practical importance in photonics and have the potential to increase communications bandwidth through spatial multiplexing. Here, we derive the coupled amplitude equations that describe the four-wave mixing (FWM) of CV beams in optical fibers. These equations allow us to determine the selection rules that govern the interconversion of CV modes in FWM processes. With these selection rules, we show that FWM conserves the total angular momentum, the sum of orbital and spin angular momenta, in the conversion of two input photons to two output photons. When applied to spontaneous FWM, the selection rules show that photon pairs can be generated in CV modes directly and can be entangled in those modes. Such quantum states of light in CV modes could benefit technologies such as quantum key distribution with satellites.
In this work we present a series of results on four-wave mixing (FWM) processes induced by structured light in atomic media. We start with a theoretical study focusing on a cascade FWM process in rubidium vapor, generating a blue light field at 420 nm. In this setting, we show that in the extended-medium regime the Hermite-Gaussian basis presents a unique property of optical structure transfer. We then exploit this result to show that, by carefully tailoring the structure of the pump beams, one can obtain desired spatial modes with cylindrical and elliptical symmetries with high fidelity at the blue light output. Following that, we consider an experimental configuration where two degenerate FWM signals are generated in different directions of space by the same pump fields. In this setup, we present results in heated rubidium vapor evidencing the simultaneous conservation of orbital angular momentum (OAM) in the two FWM processes. We show that the two-channel setting allows to encode the OAM content of the input beams onto the OAM carried by the two FWM signals. Next, we explore the transverse mode dynamics in the context of optical modes contained in the so-called orbital angular momentum Poincaré sphere (PS). Defined in analogy with a general polarization state, this family of modes is parametrized in terms of polar and azimuthal angles. We show that the two FWM signals can also be described as belonging to Poincaré spheres, and that the angles on the output PS are related to those on the input PS by well-defined symmetries. The predicted FWM intensity profiles, as well as the consequences of the symmetry properties, are in good agreement with our experimental results. We also explore interesting scenarios, such as combinations of different PS modes and the fulfillment of one of the symmetry relations in each sphere independently, and restrictions that arise in the extended-medium regime. In the last part, we study the correlations between the light fields participating in the two-channel FWM configuration. Following our recent work on cold atoms, we investigate the correlations originating from the conversion of phase-noise to amplitude-noise as a result of the light-atom interaction, and discuss our attempts to discriminate transverse spatial dependencies on these correlations. Finally, we outline the quantum theory of FWM, exploring the multi-spatial-mode nature of the generated light state and the associated spatial correlations.
Abstract Spin and orbital angular momenta of light are attractive resources to harness for encoding, and manipulating information, with applications in various quantum photonic technologies. However, to fully harness that potential, we require robust sources of high-order angular momentum photons exhibiting nonclassical correlations. Here, we propose a fiber-based source of polarization-entangled photons in high-order orbital angular momentum (OAM) modes. In our setup the pairs of photons are generated in a cylindrical fiber through a four-wave mixing process, which induces polarization, or spin entanglement. The photons are then converted to modes exhibiting large OAM by two helical gratings inscribed in the core of the fiber. We present a complete theoretical framework used to consistently describe this process, and demonstrate a robust control over the joint spectral amplitude of the generated photons.
We present a source for which multiple spontaneous four-wave mixing (SFWM) processes are supported in a few mode birefringent fiber, each process associated with a particular combination of transverse modes for the four participating waves. Within the weakly guiding regime, for which the propagation modes may be well approximated by linearly polarized (LP) modes, the departure from circular symmetry due to the fiber birefringence translates into orbital angular momentum (OAM) and parity conservation rules, i.e. reflecting elements from both azimuthal and rectangular symmetries. In our source: i) each process is group-velocity-matched so that it is, by design, nearly-factorable, and ii) the spectral separation between neighboring processes is greater than the marginal spectral width of each process. Consequently, there is a direct correspondence between the joint amplitude of each process and each of the Schmidt mode pairs of the overall two-photon state. The present paper covers work presented in Refs.1 and.2
We investigate two medium-facilitated interactions between properties of light upon propagation through optical fiber. The first is interaction between the spin and intrinsic orbital angular momentum in a linear optical medium. This interaction gives rise to fine structure in the longitudinal momenta of fiber modes and manifests in rotational beating effects. We probe those beating effects experimentally in cutback experiments, where small segments are cut from the output of a fiber to probe the evolution of both output polarization and spatial orientation, and find agreement between theoretical predictions and measured behavior. \n \nThe second is nonlinear optical interaction due to cross- and self-phase modulation between the complex-valued temporal amplitude profile of pump pulses and the amplitude profiles of generated signal and idler pulses in optical fiber photon-pair sources utilizing the four-wave mixing process named modulation instability. We develop a model including the effects of these nonlinear phase modulations (NPM) describing the time-domain wave function of the output biphoton in the low-gain regime. Assuming Gaussian temporal amplitude profiles for the pump pulse, we numerically simulate the structure of the biphoton wave function, in symmetric and asymmetric group velocity matching configurations. Comparing the overlap of the joint temporal amplitudes with and without NPM indicates how good of an approximation neglecting NPM is, and we investigate the effects of NPM on the Schmidt modes. We find that effects of NPM are small on temporally separable sources utilizing symmetric group velocity matching, but appreciably change the state of temporally entangled sources with the same group velocity matching scheme. For sources designed to produce entangled biphotons, our simulations suggest that NPM increases the Schmidt number, which may increase entanglement resource availability with utilization of a phase-sensitive detection scheme. We find that NPM effects on temporally separable sources designed with asymmetric group velocity matching produce non-negligible changes in the state structure. The purity is unaffected at perfect asymmetric group velocity matching, but if the pump is detuned from the correct wavelength, the purity degrades. The largest changes to the state due to NPM occur in long fibers with long pulse durations and low repetition rates.