Phase Matching of Orbital Angular Momentum in Rare Earth Ion Doped Solid State Systems

TL;DR

This work addresses phase matching of orbital angular momentum (OAM) in stimulated photon echoes (SPE) within rare-earth ion doped solids to enable spatial multiplexing and structured light generation. It extends SPE phase matching to include OAM, with the per-photon OAM obeying $\bm{L}_{echo}=\bm{L}_2+\bm{L}_3-\bm{L}_1$ (or equivalently $\ell_{echo}=\ell_{2}+\ell_3-\ell_1$) under the momentum constraint $\bm{k}_{echo}=\bm{k}_2+\bm{k}_3-\bm{k}_1$. Experimentally, a cryogenic $Tm^{3+}:YAG$ sample is used with three overlapping Laguerre-Gauss beams generated via an SLM and a Dammann vortex grating to encode distinct topological charges; SPEs are detected and analyzed through astigmatic transforms and spatial filtering to isolate OAM-carrying echoes. The results demonstrate controllable OAM in SPEs and show potential for optical multi-modal signal processing, quantum routing, and hyper-entangled qubits in rare-earth systems. These findings offer a pathway to OAM-enabled quantum memory and high-dimensional photonic processing in solid-state platforms without requiring complex phase stabilization.

Abstract

In this work, we demonstrate the generation of stimulated photon echos carrying unique topological charge that results from the temporal phase matching conditions of three independent beams spatially and spectrally overlapped in a cryogenically cooled rare earth ion doped solid state system. A sample of $Tm^{3+}:YAG$ was used to generalize the momentum phase matching condition to include the orbital angular momentum of the input fields. The input fields and corresponding photon echo were characterized via astigmatic transform, with results mapping directly to the expected behavior of traditional stimulated photon echos. These results demonstrate that rare earth ion doped systems are capable of spatial multiplexing, spatial filtering, and the generation of structured light, opening pathways towards real time optical multi-modal signal processing.

Figures (6)

• Figure 1: The DVG will produce a strong zero order mode as well as four diffracted orders for each input beam, each carrying a unique topological charge $\ell$. Given the initial input geometry, the equal spacing of the imaged beams will result in a single spatially overlapped beam, each carrying a different $\ell$ (represented at position [3,3]). This beam is then spatially isolated and imaged into the REI material for processing.
• Figure 2: A schematic of the free space optical system used to generate and detect SPEs with unique OAM order. All three optical inputs were first polarized in the preferential plane of the SLM before being imaged onto the center of the phase screen to generate three overlapping Laguerre-Gauss beams. These beams were then collimated before being imaged into the sample of Tm$^{3+}$:YAG contained within the CA-100. The resulting output was then collimated before being directed onto a CMOS camera for image analysis, with the camera located at the focal plane of a cylindrical lens when performing astigmatic transforms to calculate the topological charge. To isolate the SPE, the camera was triggered using the same multi-channel pulse generator that controlled the programming sequence. Definitions: HWP, Half wave plate; QWP, Quarter wave plate; SLM, spatial light modulator; CA-100, CryoAdvance 100.
• Figure 3: a.) Captured images of the modes of pulses 1, 2 and 3 as well as the SPE. Each of the pulses shows a clear central null indicative of a mode which carries OAM. The stimulated echo however shows no central null and has an Airy disk like intensity distribution. b.) Captured images of the intensity profile of each pulse and the stimulated echo at the focal plane of the cylindrical lens. After the astigmatic transform applied by the cylindrical lens, the number of nulls in each image indicates the topological charge of the incident beam. C.) A 1-D slice of each image taken along the doted line shown in the astigmatic transform images.
• Figure 4: Time domain traces showing the mode filtered echo for $\ell_1 = 0$, $\ell_2 = 0$, $\ell_3 = 0$ as well as $\ell_1 = 2$, $\ell_2 = 4$$\ell_3 = -2$. For visual clarity, both traces are normalized to the value of the maximum sample. In the case where $\ell_1 = 0$, $\ell_2 = 0$ and $\ell_3 = 0$ the echo is dominated by all of the other pulses. When the programming and probe fields are in OAM modes, they are mode filtered by the single mode fiber and are attenuated between 17 dB and 29 dB leaving the SPE as the dominant field.
• Figure 5: An array of images captured using the CMOS camera showing the intensity distribution of the SPE as a function of DVG parameters. Images are indexed according to the x and y topological charge values of the grating used to multiplex input pulses. Each image is also marked with the topological charge of the SPE, according to the parameters of the DVG.