Spiral bandwidth of four-wave mixing in Rb vapour

Laguerre-Gauss beams, and more generally the orbital angular momentum of light (OAM) provide valuable research tools for optical manipulation, processing, imaging and communication. Here we explore the high-efficiency frequency conversion of OAM in a four-wave mixing process in rubidium vapour. Conservation of the OAM in the two pump beams determines the total OAM shared by the generated light fields at 420 nm and 5.2 um - but not its distribution between them. We experimentally investigate the spiral bandwidth of the generated light modes as a function of pump OAM. A small pump OAM is transferred almost completely to the 420 nm beam. Increasing the total pump OAM broadens the OAM spectrum of the generated light, indicating OAM entanglement between the generated light fields. This clears the path to high-efficiency OAM entanglement between widely disparate wavelengths.

Laguerre-Gauss beams, and more generally the orbital angular momentum of light (OAM) provide valuable research tools for optical manipulation, processing, imaging and communication.
Here we explore the high-efficiency frequency conversion of OAM in a four-wave mixing process in rubidium vapour. Conservation of the OAM in the two pump beams determines the total OAM shared by the generated light fields at 420 nm and 5.2 µm -but not its distribution between them. We experimentally investigate the spiral bandwidth of the generated light modes as a function of pump OAM. A small pump OAM is transferred almost completely to the 420 nm beam. Increasing the total pump OAM broadens the OAM spectrum of the generated light, indicating OAM entanglement between the generated light fields. This clears the path to highefficiency OAM entanglement between widely disparate wavelengths.
Orbital angular momentum (OAM), and the transverse profile of light more generally, have the potential to greatly increase the bandwidth of both classical 1 and quantum communication [2][3][4] . In order to fully realise this potential, methods of frequency converting, manipulating and generating classical and quantum OAM states are required.
Phase-matched nonlinear processes provide the means to undertake many of these operations. The key is that these processes are phase coherent -both longitudinally and transversely -for the pump beams and generated light. As a consequence OAM, which is associated with spiral phase fronts 5 , must be conserved 6,7 , and more generally nonlinear processes can be used to manipulate transverse light modes. Frequency conversion of transverse modes 8 , images 9 and entangled OAM states 10 has been demonstrated via sum-frequency generation, whilst OAM addition has been carried out in nonlinear crystals 6,11 and atomic vapours [12][13][14] .
Where a phase coherent process generates two photons, they are highly correlated. Spontaneous parametric down conversion (SPDC) is routinely used to produce OAM entangled photon pairs 7,15 , and four-wave mixing (FWM) in atomic vapours has been used to both create 16 and store 17 entangled images. Correlations are also observed in properties other than the transverse mode. Intensity-difference squeezed beams have been generated via FWM in atomic vapours 18 , and intensity correlations have been transferred from one field to another 19 . Similar systems have been used to prepare heralded bichromatic single photons 20 , and polarisation entangled photons have been produced with cold atoms 21 .
In this work, we fully explore the transfer of OAM between fields involved in a resonantly enhanced FWM process in rubidium vapour. In our system (Fig. 1), two near-infrared pump fields (780 nm and 776 nm) generate an infrared field and a blue field [22][23][24][25] . Our results indicate that this system will be an efficient source of OAM entangled photon pairs [26][27][28] , as well as a method to add and frequency convert OAM states. This versatility is due to the very different wavelengths of the two generated fields (5.2 µm and 420 nm), which leads to the pump OAM being shared between these fields in a way that critically depends on the pump mode. We confirm this result with theoretical mode overlap models 12,29 .
To study OAM transfer in FWM we use Laguerre-Gauss (LG) pump modes, which are characterised by their azimuthal, , and radial, p, indices, with each mode carrying h of OAM 30 per photon. Conservation of OAM requires that the 5.2 µm and 420 nm fields must have the same total OAM as supplied by the pump fields, but not how the pump OAM is shared between the two generated fields as long as 780 nm + 776 nm = 420 nm + 5.2 µm .
In agreement with past observations 12-14 we find that for low values of pump OAM, the 420nm light is generated in an almost pure mode, with an OAM given by the sum of the pump modes. As the pump OAM increases ( ≥ 4), we observe the 420 nm light in an incoherent superposition of an increasing number of OAM modes, as predicted in Ref. [29], indicating that the 5.2 µm and 420 nm two-photon state becomes OAM entangled. We express this -distribution width in terms of the spiral bandwidth 31 -a measure of how many orthogonal modes could be entangled in such a state. Experimentally, we obtain the spiral bandwidth via the full and p-mode decomposition of the generated blue light for a range of pump . We find that the spiral bandwidth increases with pump OAM, but also depends on how the input OAM is divided between the two pump beams.
Previous experiments with this system have determined the OAM state of the 420 nm emission by visual inspection of an interferogram (an interference pattern formed by light overlapped with its mirror image) 12,13 . Here we perform a full Fourier analysis of the interferogram to precisely identify the modal superposition. We show that our experimental observations agree with a detailed FWM model that incorporates a full ( and p) LG p modal description of the four fields.

Results
The experimental setup and theory of the generated light LG p decomposition are detailed in the Methods.
Mode decomposition via Fourier analysis. Many methods to quantitatively measure the OAM spectrum of a beam or even a single photon have been demonstrated, including fork diffraction gratings used as filters followed by on-axis detection 7,32-34 , cascading Mach-Zehnder interferometers containing Dove prisms 35,36 and transformation optics 37,38 . All methods are applicable to both coherent and incoherent superpositions of OAM ( ) modes, however the identification of p modes proves more challenging. The latter two methods above give no or only indirect information on p modes. Cycling through the relevant fork gratings does, in principle, allow the identification of the complete mode decomposition, but is time-consuming and requires an additional SLM.
Here we use an experimentally simple method, based on Fourier analysis of an interferogram of the light mode of interest, obtained using a Dove prism interferometer ( Fig. 1). Unlike previous experiments with this FWM system 12,13 , where the 420 nm mode was determined by visual inspection of interference patterns, we use Fourier analysis to precisely extract the full incoherent and pmode decomposition from each interferogram. We note that this method is suitable only for classical light beams.
For a pure LG p mode, the interferogram has 2 azimuthal lobes 39 . While for a coherent superposition we would expect the fields to add, for an incoherent superposition the intensities add, and the interferogram shows the sum of the constituent mode interference patterns: where P p is the relative power in each mode, R p is the radial intensity profile of each mode and φ p is the modedependent interferometer phase. Note for = 0 modes the total interferogram amplitude entirely depends on interferometer phase, φ 0 p , and thus the relative power in = 0 modes cannot be found from the interferogram. To allow = 0 mode measurement, and correct for slight discrepancies in 50:50 beam splitter transmission, we take three images at the interferometer output: the interferogram, I T , as well as the intensity profiles in arms A and B, I A and I B (Fig. 2 (a)). From theses images we calculate a corrected interferogram I C (r, θ) ( Fig. 2 (b)): where I AB (r) = I A (r) + I B (r) /2, with I A (r) and I B (r) the average radial profile of I A and I B , respectively. For further details see Supplement 1. The form of Eq. 1 means that the corrected interferogram is readily Fourier analysed. Obtaining the azimuthal profile by integrating I(r, θ) ( Fig. 2 (c)), then performing a one-dimensional Fourier transform, results in a Fourier spectrum where 2 terms give P | | = p P | | p , for | | > 0. Our current method is limited to measuring | |, but can be inferred from OAM conservation 13 .
Unlike the -decomposition, which arises from the rotational symmetry of the beam, the p-decomposition is not uniquely defined, but depends on the assumed waist. We choose a beam waist such that one 'target' mode dominates. We find the p-decomposition for each by separating the corrected interferogram into -components using two-dimensional Fourier filtering. The radial profile of each component is then fitted with an incoherent superposition of p-modes (with relevant ). Combining the information from the and p decompositions, the relative power in all modes with | | > 0 is found. To get the = 0 modes a final fit is performed to the radial profile of the corrected interferogram ( Fig. 2  (d)). The model for the fit is the total | | > 0 radial profile, multiplied by a single scale factor, plus an incoherent = 0 mode sum. To verify our analysis we use the mode decomposition results to calculate a reconstructed angular and radial profile (Fig. 2 (c) and (d)), finding good agreement with the measurements.
For later comparison with the generated 420 nm light later, Fig. 3 (a) shows the beam profiles, I A , and uncorrected interferograms, I T , for LG 776 0 pump modes ( 776 = 0 → 8) with the resulting mode decompositions in Fig. 3 (b). Only p = 0 modes are shown, as they constitute > 96% of the power in each case, however the full analysis up to p = 3 is in Supplement 1, upper Fig. S1. The corresponding power in the target mode, P t = P 776 p=0 is shown in Fig. 3 (c). The measured mode purity is high, P t > 0.97 for 776 < 5 with a slight decrease as 776 increases. The 780 nm pump beam has similar mode purities (Supplement 1, lower Fig. S1).
Spiral bandwidth broadening. We now consider the FWM experimental results. In the first experiment only the 776 nm pump beam carries OAM, with the 780 nm beam in the Gaussian LG 0 0 mode. We measure the 420 nm mode decomposition P B p B up to B = 8. The 780 nm beam focus is axially offset to before that of the 776 nm beam to improve the spatial overlap of the fields (Methods). The intensity profiles, I A , and uncorrected interferograms, I T , of the generated 420 nm beams for each pump mode are shown in Fig. 4 (a).
Mode decomposition was carried out for each interferogram, considering LG 0≤ ≤10 0≤p≤3 modes. We find that < 10% of the light is generated in p > 0 modes (< 2% theoretically), and present only p = 0 results here. The experimental and theoretical mode decompositions are in Fig. 4 (b) and (c), respectively (see Supplement 1, upper Fig. S2 for p > 0 results). Fig. 4 (e) shows the relative power in the 420 nm target mode, P t = P 776 p=0 . For low pump modes, 776 ≤ 3, the blue light is generated in a single mode, indicated by the high visibility fringes in the interferogram. In this regime, the results are consistent with nearly all of the pump OAM being transferred to the 420 nm emission. The 776 nm OAM state is efficiently frequency converted to 420 nm, with measured mode purity higher than the theoretical prediction. The FWM gain may be sufficient to produce a 'lasing' effect, amplifying the optimal mode 43 .
The asymmetry in the theoretical generated field OAM mode distributions is due to their large wavelengths difference, i.e. 420 nm and 5.2 µm (Fig. 4 (c) and (d)), and therefore very different waists (Eq. 4). The waist of the 5.2 µm light is 2.6 times that of the pump light, and so for low 776 there is much better overlap between the fields if the 5.2 µm field is in the Gaussian = 0 mode 12 (Fig. 4 (d) -see Supplement 1, Fig. S2 for p > 0 mode powers).
As 776 increases, the 420 nm light spreads over an increasing number of modes -apparent in the decreasing fringe visibility at the top and bottom of the interferograms ( Fig. 4 (a)). The position of high visibility is determined by the interferometer geometry and in particular the Dove prism orientation which causes the extra reflection. Our reflection takes place about the horizontal axis where nearby parts of the beam interfere, therefore each separate mode's interference pattern is necessarily in phase along the horizontal axis.
When the 420 nm light is observed in a range of modes, due to OAM conservation, the 5.2 µm field can no longer be solely generated in the LG 0 0 mode. The radius of an LG mode increases with √ so as 776 increases the overlap with the higher order 5.2 µm modes improves. The two-photon 5.2 µm and 420 nm field is then generated as a coherent superposition of different combinations of OAM-conserving modes. The 776 ≥ 4 results, with power spread over more than one 420 nm mode, strongly indicate that generated 5.2 µm and 420 nm photon pairs are OAM-entangled, although further experiments are required to verify this. The number of modes involved in the entangled state, the spiral bandwidth, depends on the width of the 420 nm -decomposition, ∆ = 2 − 2 , p = 0. We observe a strongly mode-dependent spiral bandwidth, that increases with increasing pump OAM (Fig. 4 (f)). Finally, we note the effect of Gouy phase matching on the mode decomposition. Both the theoretical and experimental results show a 420 nm -decomposition asymmetry; there is relatively more power in modes with lower than that of the target mode. To conserve OAM, if the 420 nm light has > 776 , then the 5.2 µm mode must have negative . Considering Eq. 7 this situation cannot be well Gouy phase matched (since p ≥ 0), and therefore modes with > 776 are less likely. With axially offset 780 nm and 776 nm foci, the 780 nm Gouy phase is essentially constant through the FWM region. This is included in our theoretical model (Methods), and reduces the effect of Gouy phase matching compared to when both beams are confocal in the following section.
Shared pump OAM. Naïvely one might think it doesn't matter whether the OAM is provided by only one or both of the pump beams. However, we observe a reduction of spiral bandwidth for a given target mode if both pump beams carry OAM. In our second FWM experiment the two pump beams are shaped into the same LG mode, 780 = 776 , p 780 = p 776 = 0, overlapped and focused into the centre of the rubidium cell. Fig. 5 (a) shows the intensity profile, I A , and uncorrected interferogram, I T , of the generated 420 nm beam for each pump , whilst Fig. 5 (b) and (c) show the experimental and theoretical p = 0 mode decompositions (see Supplement 1, lower Fig. S2 for p > 0). Like Fig. 4, for low pump beams, the 420 nm mode decomposition is consistent with all OAM transferred to the 420 nm light, and target mode purity P t = P 776 + 780 p=0 higher than predicted by theory. Here, the blue light OAM is the sum of the two pump beam OAM, and we demonstrate efficient OAM addition. As the pump increases we see the 420 nm power spread over more modes and the available spiral bandwidth increases (Fig. 5 (f)), both in the experiment and the theory.
A comparison of providing OAM from one or both pump beams can be seen in Figs. 4 and 5, where for example the identical target mode LG 8 0 is highlighted by boxes in both cases. We observe a larger spread of 420 nm modes, indicating a larger spiral bandwidth, when all of the OAM is provided by the 776 nm field (Fig. 4) than when the pump OAM is shared equally between the two pump fields (Fig. 5). This is theoretically expected due to increased mode overlap for 780 = 776 , however experimentally we observe a larger discrepancy. The difference may be explained by our model giving a poor prediction of the 5.2 µm and 420 nm waists (see Supplement 1, Fig. S5 for further details). Comparing Figs. 4 and 5 also shows the relative importance of Gouy phase matching: in Fig. 4 and 5 the pump beams have foci that are axially displaced and co-axial, respectively. The increased pump beam propagation symmetry in the latter case leads to stronger Gouy phase matching. As a result, both experimentally and theoretically, there is stronger asymmetry in the 420 nm -distribution about the target mode in Fig. 5 Fig. 4. This is also apparent in the predicted 5.2 µm mode decomposition, where essentially no light is predicted with < 0 in Fig.  5, but some is predicted in Fig. 4.

Discussion
We have developed a method of precisely determining the decomposition of incoherent beam modes and used this to analyse the transverse light created via FWM in Rb vapour. Due to the large wavelength difference of the two generated fields, the OAM distribution between them is pump-mode dependent. A simple theoretical model 12,29 , which we adapted to include a full and p description of all four fields, describes our system well, but could be improved by including propagation effects and a more accurate generated field waist prediction.
For small pump OAM ( < 4), our results are consistent with all OAM being transferred to the 420 nm light. In this regime the FWM process can be faithfully used to both frequency convert and add OAM states. As the pump OAM increases the 420 nm light is generated in an incoherent superposition of an increasing number of modes. This indicates that the pump OAM is shared between the 5.2 µm and 420 nm light, which is likely to lead to highly efficient generation of OAM entangled photon pairs. We have inferred the available spiral bandwidth and find that it increases with increasing pump OAM.
In addition, the OAM distribution between the generated fields depends on how the pump OAM is supplied. We see a significant decrease in spiral bandwidth when the pump OAM is shared equally between the pump beams rather than being supplied by only one pump beam. Our results also show the importance of Gouy phase matching, which results here in an asymmetricdistribution about the target mode.
As well as demonstrating a means of manipulating and generating OAM states, these results are relevant for schemes involving inscription and storage of phase information in atomic gases. Although we currently work with an atomic vapour, the techniques presented here are well-suited for use with cold atoms 40,41 , enabling long lifetime high-dimensional quantum memories with wavelength versatility. Finally, we note the 420 nm output can be constrained to a single transverse mode, even for high input OAM, if a cavity is added to the system. The output is then determined by a combination of the coherent FWM pumping mechanism and the cavity modes.
The datasets used in this work are available online 42 .

Methods
Energy levels and wavelengths. Two near-infrared pump fields (780 nm and 776 nm) generate an infrared field (5.2 µm) and a blue field (420 nm). With Gaussian pump beams, this resonantly enhanced process can be carried out very efficiently [22][23][24][25] , generating mW levels of 420 nm light with low power pump beams 25 . Although we use single pass FWM here, the process can also be enhanced through the use of a cavity resonant with either the generated 43 or pump light 44 . Laser locks. In order to minimise single-photon absorption 25 , the 780 nm laser 45 is locked roughly half way between the hyperfine states of the 85 Rb 5S 1/2 ground state (+1.6 GHz from the 5S 1/2 F = 3 to 5P 3/2 F = 4 transition). The 776 nm laser is then locked two-photon resonant with the 85 Rb 5S 1/2 F = 3 to 5D 5/2 F = 5 transition. The two generated fields are quasi-resonant with the downward cascade via the 6P 3/2 state 12,43,46 .
The optical setup. The two pump fields are shaped independently (Fig. 1) into a range of LG modes by displaying the corresponding holograms each on one half of a spatial light modulator (SLM, Hamamatsu LCOS-SLM X13138). The required phase holograms are calculated following method C outlined in Refs. [47,48], and first proposed in Ref. [49]. The shaped pump beams are combined on a non-polarising beam splitter (NPBS), circularly polarised, and focused to a w 0 = 25 µm waist at the centre of aa 120 • C Rb cell (P Rb ∼ 9 × 10 −4 mbar). The Rayleigh range z R = 2.5 mm is much less than the 25 mm cell length. FWM generates light at 420 nm and also 5.2 µm. The 5.2 µm field is absorbed here by the glass of the cell, however it can be observed from sapphire cells 22,50 . Mode decomposition. The transverse mode decomposition of the pump and generated fields are determined from interferograms obtained by overlapping a beam with its mirror image. A single field is chosen with a spectral filter (for the 420 nm field) or by blocking one of the pump beams, and the light is then linearly polarised. The light passes through a Mach-Zehnder interferometer, with a Dove prism in one of its arms to generate the light mode's mirror image. The resulting interference pattern after the final NPBS contains azimuthal lobes, when carefully aligned. Interferogram analysis gives the and p-mode decomposition of each of the fields. See Supplement 1 for further experimental details. Theory: LG modes. The generated field and p mode decomposition for a particular pair of pump modes can be predicted by considering the FWM field overlap within the Rb cell 12,29 . The LG mode electric field is given in cylindrical polars by: LG p is an associated Laguerre polynomial, w = w 0 1 + (z/z R ) 2 is the beam radius for a waist w 0 , the Rayleigh range is z R = πw 2 0 /λ, and the spherical phasefronts and Guoy phase are, respectively, Φ S = kr 2 z/(2(z 2 + z 2 R )) and Φ G = −(2p + | | + 1) arctan(z/z R ). Theory: Boyd criterion and Guoy phase. In order to have efficient wave mixing the Rayleigh range of the four fields must be matched -known for Gaussian beams as the Boyd criterion 51 . We make the assumption that this holds for higher order transverse modes, and thus the 5.2 µm and 420 nm field waists satisfy: The probability amplitude to generate a particular pair of 5.2 µm and 420 nm modes, when pumping with LG 780 p 780 and LG 776 p 776 , can be found by evaluating the overlap integral where the mode waists are given by Eq. 4 and L is the cell length 12,29 . The azimuthal integral ensures OAM ( ), is conserved whilst the radial integral considers the field spatial overlap. The z integral specifies the beam propagation through the cell. The Gouy phase, Φ G in Eq. 3, describes a modification of the phase of a focused beam compared to that of a collimated beam. In order for phase matching to be maintained between the pump and generated beams throughout the cell, they must have identical Gouy phasesa requirement we call Gouy phase matching. This means that the mode order 2p + | | must be conserved, i.e. for p = 0 pump modes, the only Gouy-phase matched generated modes satisfy: Theory: generated fields. We assume that the 5.2 µm and 420 nm light is generated as a coherent superposition of two-photon states, each weighted by coefficients c B , IR Although the two-photon field is coherent, the 420 nm field alone is an incoherent mixture of modes. We obtain the probability of observing the 420 nm light in a specific mode by summing over all . The model neglects the propagation effects of absorption and Kerr lensing -a reasonable assumption as the 780 nm laser is detuned +1.6 GHz and -1.4 GHz from the F = 3 and F = 2 ground states, respectively 25 .
In the experiment, we compare FWM under two conditions: all pump OAM carried only by the 776 nm field (with the 780 nm field in the LG 0 0 mode); and with the OAM shared evenly between the pump beams. In the first case we axially offset the 780 nm field focus to z off = 9.6 mm (3.8z R ) before that of the 776 nm field, to improve the spatial overlap of the higher order 776 nm modes with the Gaussian 780 nm beam. This is included in the model by performing the transformation z → z + z off on the 780 nm mode before evaluating Eq. 5. This causes a mismatch in the pump beam phase front curvature, but we assume overall phase front matching of the two pump fields with the generated two-photon field. This is in agreement with our experimental observations; we observe a change in the 420 nm field collimation when the 780 nm focus shifts.