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Optics Letters

Suitable state bases for nonlinear optical mode conversion protocols

D. G. Pires, J. C. A. Rocha, Alcenísio J. Jesus‐Silva, E. J. S. Fonseca

Published Jun 22, 2020• 7 citations
Publisher

Abstract

By considering parity-defined Laguerre-Gaussian (LG) and Hermite-Gaussian (HG) beams as input modes, we present arguments through experimental and theoretical results in order to affirm that using HG modes as bases is more suitable for optical mode conversion than using LG modes. By analyzing the normalized overlap integral and the generated modes, we determine a clear rule for the dominant mode for nonlinear mixing of HG beams, while the same is not possible for LG beams. In addition, examples of optical modal conversion using both HG and LG modes as input beams are demonstrated.

DOI
10.1364/ol.394640

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By considering parity-defined Laguerre-Gaussian (LG) and Hermite-Gaussian (HG) beams as input modes, we present arguments through experimental and theoretical results in order to affirm that using HG modes as bases is more suitable for optical mode conversion than using LG modes. By analyzing the normalized overlap integral and the generated modes, we determine a clear rule for the dominant mode for nonlinear mixing of HG beams, while the same is not possible for LG beams. In addition, examples of optical modal conversion using both HG and LG modes as input beams are demonstrated.
Abstract Precise light manipulation in optical media has been important for developing new communication protocols, particle tracking and control, environmental sensing and quantum computing, to cite a few applications. Here, we explore the nonlinear light-matter interactions of Hermite-Gaussian (HG) beams with a rubidium vapor. We analytically and numerically predict the possibility of generating a nearly pure optical HG mode as a result of the four-wave mixing of two different HG modes. In contrast, four-wave mixing of other linear solutions of the paraxial wave equation in a third-order nonlinear medium results in the up-converted beam consisting of a superposition of multiple modes. In addition, we identify a large family of solutions of the paraxial wave equation, including cylindrical and elliptical modes, that can be generated in the process of the four-wave mixing by precisely adjusting the weights of the input HG beams. This study may find applications in the development of new classical and quantum optical sources, with more spatial versatility.
Optical orbital angular momentum transformation and corresponding azimuthal-mode selection rules have been studied exhaustively for various nonlinear optical interactions. However, nonlinear transformation of the radial mode has not been systematically studied since the pioneering work [Phys. Rev. A 56, 4193 (1997)]. In this paper, we theoretically study and experimentally verify the radial modal transitions of Laguerre-Gauss (LG) modes in parametric up-conversion. Specifically, we provide a general solution that describes the sum-frequency generation (SFG) field excited by two arbitrary LG modes. Based on the solution, one can predict the full spatial complex amplitude of SFG fields upon propagation precisely and readily obtain the associated full-field selection rule including both azimuthal and radial modes. This paper provides a theoretical basis for quantum and nonlinear optical research involving parametric up-conversion of complex structured light and paves the way for future work on full-field transformation of spatial modes in other nonlinear interactions.
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Recent developments in laser physics have called renewed attention to Laguerre–Gaussian (LG) beams of paraxial light. In this paper we consider the corresponding LG modes for the two-dimensional harmonic oscillator, which appear in the transversal plane at the laser beam's waist. We see how they arise as Wigner transforms of Hermite–Gaussian modes, and we proceed to find a closed form for their own Wigner transforms, providing an alternative to the methods of Simon and Agarwal. Our main observation is that the Wigner transform intertwines the creation and annihilation operators for the two classes of modes.
Photoexcitation of trapped ions by Hermite–Gaussian (HG) modes from guided beam structures is proposed and investigated theoretically. In particular, simple analytical expressions for the matrix elements of induced atomic transitions are derived that depend both on the parameters of HG beams and on the geometry of an experiment. By using these general expressions, the (Formula presented.) electric octupole (E3) transition is investigated in an Yb+ ion, localized in the low–intensity center of the HG10 and HG01 beams. It is shown how the corresponding Rabi frequency can be enhanced by properly choosing the polarization of incident light and the orientation of an external magnetic field, which defines the quantization axis of a target ion. The calculations, performed for experimentally feasible beam parameters, indicate that the achieved Rabi frequencies can be comparable or even higher than those observed for the conventional Laguerre–Gaussian (LG) modes. Since HG-like modes can be relatively straightforwardly generated with high purity and stability from integrated photonics, these results suggest that they may form a novel tool for investigating highly-forbidden atomic transitions.
Photoexcitation of trapped ions by Hermite–Gaussian (HG) modes from guided beam structures is proposed and investigated theoretically. In particular, simple analytical expressions for the matrix elements of induced atomic transitions are derived that depend both on the parameters of HG beams and on the geometry of an experiment. By using these general expressions, the (Formula presented.) electric octupole (E3) transition is investigated in an Yb+ ion, localized in the low–intensity center of the HG10 and HG01 beams. It is shown how the corresponding Rabi frequency can be enhanced by properly choosing the polarization of incident light and the orientation of an external magnetic field, which defines the quantization axis of a target ion. The calculations, performed for experimentally feasible beam parameters, indicate that the achieved Rabi frequencies can be comparable or even higher than those observed for the conventional Laguerre–Gaussian (LG) modes. Since HG-like modes can be relatively straightforwardly generated with high purity and stability from integrated photonics, these results suggest that they may form a novel tool for investigating highly-forbidden atomic transitions.