The angular momentum of light has been the subject of fundamental discussions about its analogies with atomic variables for nearly one century [1]. In 1993, Beijersbergen and colleagues demonstrated that, indeed, laser light can carry orbital angular momentum (OAM) by means of an appropriate preparation [2]. Thereafter, the OAM of light has been employed for a vast quantity of classical applications covering from microscopy and micro-manipulation to astrophysics and medicine [3]. This variable became an additional resource of quantum engineering once researchers were able to transfer it to correlated photons through spontaneous parametric down conversion (SPDC) [4, 5]; large Hilbert spaces became available for the subsequent treatment of entangled-photon pairs [6–8].

Since OAM can be transferred to and retrieved from atoms [9], it is also possible to generate quantum light-carrying OAM by inducing four-wave mixing (FWM). Controlling the electromagnetic degrees of freedom through this non-linear process makes it possible to generate light correlated in the time [10], useful to build, for example, quantum memories [11]. Ladder-type schemes of FWM in the alkali elements allow to convert light from one end to the other end of the optical spectrum, and even beyond. For example, exciting applications arose from generating and detecting electromagnetic fields at terahertz frequencies through FWM [12–15]. Here, we drive attention to the double transition 5S_1/2 → 5P_3/2 → 5D_5/2 of Rb⁸⁷, which yields a beam at 420 nm during the second step of its cascade decay that can be readily detected [12] (Figure 1). This scheme has been a workhorse to those interested in developing the usage of OAM on light with atomic origin because the up-converted beam is relatively simple to collimate and optimize in setups where FWM is induced on hot atoms [16, 17]. Thus, the experiments involving this collimated blue light (CBL) are suitable for building scalable and robust devices, as desirable for practical applications. Quantum technologies are included since phase matching during the FWM process indicates that the OAM entanglement should be present between the CBL and the electromagnetic field at 5 μm (CMW), emitted during the cascade decay, whenever the appropriate choice in the parameters of the pump beams is carried out, especially if the total topological charge of the pump beams is large enough [18].

Single, optical vortices were first up-converted by [19]. In [20], it was shown that it is possible to perform arithmetic operations with the OAM traveling on both pump beams through FWM. Moreover, optical vortices with a helicity up to ±30 are transferable to the CBL as well [21]. In those experiments, the FWM process was pumped with Laguerre–Gauss beams that carry one phase singularity along a straight dislocation line. However, since 2002, it has been possible to generate arrays of optical vortices on laser light using Mathieu modes [22], members of the quasi-propagation invariant beam family [23]. This family of structured beams has given birth to numerous scientific discoveries through micro-manipulation of biological materials [see for example [24]]. More recently, they have yielded new methods to control the spatial and the correlation properties of photon pairs generated by SPDC [25].

In a recent publication, we reported the up-conversion of quasi-propagation invariant Mathieu beams through an FWM process in hot atoms [26]. There, the light-mode analysis was performed by studying in detail the Fourier and configuration spaces of the electromagnetic fields. These are well-established methods for experiments using non-linear crystals. However, they were introduced to the context of atomic gases in [26]. For those experiments neither the pump beams nor the CBL exhibited vortices. In this article, we report that modes containing arrays of optical vortices are also inherited via FWM. We also show that Michelson–Morley interferometry is a suitable tool to confirm this fact. A simulation of the classical interference pattern serves as a reference to locate the phase singularities. We induced FWM as depicted in Figure 1: with a Gaussian beam (G) resonant to the 5S _1/2 → 5P _3/2 transition and a helical Mathieu–Gauss beam (HM-G) exciting the 5P _3/2 → 5D _5/2 second step. We demonstrated that a non-trivial density of OAM was transferred from the pumping HM-G beam to the generated CBL. Our findings complement studies of non-linear processes, where the local density of angular momentum of light is connected to a spatially dependent polarization [27]. The results that we present here add two tools to experiments of this kind: conversion of light with a quasi-propagation invariant structure and several dislocation lines; and an extraordinary control over the properties of the generated light based on manipulating the atomic states involved in the non-linear process.

In this section, we describe a few basic features of the structured light fields used in the experiment. We start with the elementary Mathieu modes that do not exhibit vortices but are the basis from which helical Mathieu beams—which may have more than one dislocation line—are built.

Elementary Mathieu modes are written in terms of scalar functions Ψ ( r ⃗ , t ) that solve the following wave equation: ∇ 2 − 1 c 2 ∂ 2 ∂ t 2 Ψ r ⃗ , t = 0 , (1) in elliptic-cylindrical coordinates {ξ, η, z}. These coordinates are related to the Cartesian space {x, y, z} by the transformations: x + i y = h ⁡ cosh ξ + i η , ξ ϵ 0 , ∞ , η ϵ 0,2 π ; and z = z , (2) where the constant h corresponds to half the inter-focal distance of the ellipses defining the coordinate system. The wave-equation in elliptic-cylindrical coordinates can be solved by the separation of variable method. By demanding the solutions to have a well-defined parity with respect to reflections through the z-plane, and to take finite value as ξ → ∞, the following expressions for the scalar wave function Ψ ( r ⃗ , t ) are found [28]: Ψ κ e r ⃗ , t = A κ J e n ξ , q c e n η , q e i k z z − ω t , Ψ κ o r ⃗ , t = B κ J o n ξ , q s e n η , q e i k z z − ω t . (3) Here, ce _n (η, q) and se _n (η, q) are the real even and odd ordinary solutions of the Mathieu equation: d 2 d η 2 + a n − 2 q ⁡ cos ⁡ 2 ⁡ η f n p η , q = 0 , q = h κ ⊥ 2 2 , κ ⊥ 2 = ω / c 2 − k z 2 ; (4) and Je _n (η, q) and Jo _n (η, q) solve its modified analog as follows: d 2 d ξ 2 + a n − 2 q ⁡ cosh ⁡ 2 ⁡ ξ F n p ξ , q = 0 . (5) In general, the characteristic values a _n and b _n , for even and odd Mathieu functions, respectively, are ordered by the progressive parameter n. For a given n, a _n ≠ b _n . The label κ in Eq. 3 denotes a set of separation constants ω, k _z , n.

The wave function Ψ ( r ⃗ , t ) satisfies the following eigenvalue equations [29], i ∂ ∂ t Ψ r ⃗ , t = ω Ψ r ⃗ , t , − i ∂ ∂ z Ψ r ⃗ , t = k z Ψ r ⃗ , t , (6a) 1 2 l ⃗ ̂ + z l ⃗ ̂ − z + l ⃗ ̂ − z l ⃗ ̂ + z Ψ r ⃗ , t = a n Ψ r ⃗ , t , l ⃗ ̂ ± = r ⃗ − 0,0 , ± h × − i ∇ ⃗ ; (6b) ( l ̂ ± ) z is the z–component of the l ⃗ ̂ ± vector.

As a consequence, once the generalization to vectorial electromagnetic waves is carried out and the standard quantization is performed [30], in the quantum realm, the parameters {ω, k _z , a _n } are assigned to a photon described by these EM modes, an energy ℏω, a linear momentum along the main propagation axis ℏk _z , and an algebraic mean value of the z-component of the angular momentum with respect to the axis passing through the foci of the elliptic coordinate system, ℏ ² a _n .

The angular spectrum of Mathieu fields, S n p θ k ⃗ , φ k ⃗ ; a , q = f n p φ k ⃗ , q δ sin θ k ⃗ − κ ⊥ c / ω | cos θ k ⃗ | sin θ k ⃗ , (7) allows them to be written as the superposition of plane waves, Ψ p r ⃗ , t = ∫ d ω α ω ∫ d 3 k δ ω − c | k ⃗ | e i k ⃗ ⋅ r ⃗ − i ω t S p θ k ⃗ , φ k ⃗ . (8) The delta factor in Eq. 7 guarantees cylindrical symmetry on the field by restricting the participating plane waves to those sharing a common modulus κ _⊥ of their transverse wave vector. Nevertheless, actual realizations of Mathieu beams involve a conic-shell volume within the wave-vector space derived from replacing the delta distribution by a properly normalized Gaussian distribution [31]: δ sin ⁡ θ − κ ⊥ c / ω → 1 c σ / ω 2 π e − sin ⁡ θ − κ ⊥ c / ω 2 2 c σ / ω 2 (9) with waist σ. This spectrum can be codified in a spatial light modulator (SLM) to generate electromagnetic Mathieu modes in the paraxial regime (κ _⊥≪|k _z |) with a polarization determined by the quasi-plane waves that impinge the SLM. Transfer of the elementary Mathieu modes described here from infrared to blue light via FWM in atomic gases has recently been reported [26].

Optical vortices in scalar fields are locations where the phase is not well defined and exhibits a change of 2mϕ along any closed loop around them; the integer m is called topological charge. Close to a vortex, the field magnitude is zero, but the density of orbital angular momentum is not null. Even though the phase structure of scalar solutions Ψ κ ( e ) ( r ⃗ , t ) and Ψ κ ( o ) ( r ⃗ , t ) do not exhibit this kind of singularity, a proper superposition of them can give rise to one or more optical vortices with topological charges ±1. This happens for generalized-helical Mathieu beams with angular spectrum S κ h θ k ⃗ , φ k ⃗ = I n e c e n φ k ⃗ , q + i I n o s e n φ k ⃗ , q . (10) Most studies in the literature consider the case for which the real constants I n ( e ) and I n ( o ) have the same absolute value [32].

By varying the plane of observation, optical vortices create the so-called dislocation lines. In the ideal case (σ → 0), helical Mathieu beams are propagation invariant, and the dislocation lines are straight and parallel to the main direction of propagation. For actual helical Mathieu–Gauss beams, the dislocation lines are open but exhibit a slight curvature due to the unavoidable partial focusing of any Gaussian-like beam.

For the experiments reported here, we used a helical Mathieu–Gauss mode with n = 4, I n ( e ) = I n ( o ) , κ _⊥ = 11 mm⁻¹, σ = 0.038κ _⊥, and q = 21.78 as an illustrative example. A simulation of its intensity profile on the z = 0 plane is shown in Figure 2A; some vortices are highlighted with blue circles where the intensity is null. The corresponding phase diagram is shown in Figure 2A. There, one can corroborate that the phase changes by 2π around the neighborhood highlighted in (A), so that the topological charge is unitary. Figures 2C,D show this beam in the Fourier space. Figure 2C depicts its intensity in the k _x − k _y plane, and Figure 2D plots its angular dependence along the central κ _⊥ circle.

To perform experiments, we used a similar apparatus as that reported in [26] with an additional interferometric module. Here, we describe only its main features and the Michelson–Morley arrangement by which the optical singularities were detected.

The experimental setup is schematized in Figure 3. Both HM-G and G pump beams are frequency-stabilized in separate spectroscopy setups which are not shown. Every data reported in this article were taken with G resonant to the 5S_1/2 → 5P_3/2 transition (δ ₁ = 0); δ ₂ was set to −16.2 MHz, optimizing the intensity of CBL on its Fourier plane within a ±20 MHz range. The top part of Figure 3A illustrates the optical arrangement for generating arbitrary Mathieu–Gauss beams with a phase-only SLM [33]. Their cross section has a long diameter of about 4 mm. This size is matched to the G beam with the help of telescope T1, as shown in the bottom part of Figure 3A. Experiments were performed by saturating the first FWM step with a power of 27 mW on G; HM-G carried 7 mW only. Both beams are overlapped on an interference filter F1 to co-propagate them across a heated spectroscopy cell. This guarantees the Boyd criteria for the efficiency in the FWM process The Fourier space of HM-G is imaged right before interacting with atoms by placing CMOS-I at the focal plane of lens L1. The CBL is equally monitored by focusing it with L2 on CMOS-2.

Figure 3B depicts the Michelson–Morley module added to analyze the phase of CBL. This interferometer is a variation of the simplest technique to detect an optical vortex: to interfere with the studied beam with an inclined plane wave, resulting in a fork-like interferogram. By counting the fork number in the resulting pattern and observing their relative orientation, the vortex order and its corresponding sign can be precisely assigned. We did not try to produce a blue plane wave to interfere with CBL because that requires an extra laser. Instead, we split the CBL, letting it to interfere with itself. Clear inference patterns can be readily observed by overlapping the center of one arm with the external field of the other, where the CBL-phase structure has its smallest variations [34].

The Michelson–Morley module is placed instead of lens L2 and CMOS-2 at the right end of Figure 3A—right after the interference filter IF2, which removes remnants of pumping light. Two arms with CBL are created by a 50:50 beam splitter cube. Each one of them is retro-reflected by its respective mirror (M1 and M2 in Figure 3B) in order to overlap them on a simple CMOS camera. For this interferometer to work, the arms should be slightly misaligned. The distance between them when impinging on the CMOS chip is controlled using a translation stage driving the angle θ of M1.

We characterized both electromagnetic fields by measuring their angular spectra from images of their Fourier plane to show that the structure of HM-G is transferred to CBL through the FWM process. This allows to formulate a compact expression for the two beams in terms of parameters that define the ideal Mathieu mode programmed to the SLM [26]. The analyses were performed for the illustrative example described in Figure 2. Its relevant parameters are shown in Table1. The interference patterns of CBL with itself confirmed that the full set of vortices, theoretically expected on the major axis of the Mathieu mode, are present in the generated light.

We demonstrated that non-trivial phase structures, composed of arrays of optical vortices, are transferable through FWM in atomic gases using a ladder-type double transition—that can up- and down-convert light. This was carried out by, respectively, exciting its first and second steps with Gaussian and Mathieu–Gauss beams. We confirmed that the HM-G pump beam was appropriately characterized by the parameters of an ideal helical Mathieu mode, as well as the generated CBL, by measuring their angular spectra in their respective Fourier planes. For probing phase singularities, we successfully introduced Michelson–Morley interferometry to the analysis of light generated by FWM in atomic gases.

Measurements in the Fourier space showed that the transverse components κ ⊥ G − M ≈ κ ⊥ CBL . Therefore, momentum perpendicular to the propagation of helical Mathieu modes is fully transferred from HM-G to CBL as well as happens individually for the odd and even cases [26]. During the same analysis, we showed that the ellipticity parameter q of the Mathieu modes is also transferred within our experimental accuracy. With Michelson–Morley interferometry, we verified that the zero-field regions on the semi-major axis of CBL carry the optical vortices of the mode that was fed to atoms by HM-G. We also observed that the topological charge of vortices on HM-G is equal to the topological charge of vortices on CBL. This testifies that the microwave field CMW should not be carrying OAM in this very context. Nevertheless, this situation should change if the pump beam G is imprinted with phase structure as well.

Our work is a step forward to classical applications requiring multiple vortices on light with frequencies hard to achieve such as free-space multiplexing with OAM [38]. In principle, our results can be extended to generate twin beams controllably carrying multiple phase singularities. Therefore, they also contribute to enhancing OAM-multiplexing with quantum light. This exciting application has been recently demonstrated [39]. It has been subsequently employed to perform OAM quantum teleportation, tripartite entanglement, and quantum dense coding in photon pairs with similar frequencies [40, 41, 42]. Therefore, our contribution may well be the key that opens access toward implementing these applications with differently colored, quantum-correlated light carrying several OAM channels in addition to remote preparation of highly structured optical states [27].