Fast generation and detection of spatial modes of light using an acousto-optic modulator

Abstract
Spatial modes of light provide a high-dimensional space that can be used to encode both classical and quantum information. Current approaches for dynamically generating and measuring these modes are slow, due to the need to reconfigure a high-resolution phase mask such as a spatial light modulator or digital micromirror device. The process of updating the spatial mode of light can be greatly accelerated by multiplexing a set of static phase masks with a fast, image-preserving optical switch, such as an acousto-optic modulator (AOM). We experimentally realize this approach, using a double-pass AOM to generate one of five orbital angular momentum states with a switching rate of up to 500 kHz. We then apply this system to perform fast quantum state tomography of spatial modes of light in a 2-dimensional Hilbert space by projecting the unknown state onto six spatial modes comprising three mutually unbiased bases. We are able to reconstruct arbitrary states in under 1 ms with an average fidelity of 96.9%.

Introduction
The application of spatial of modes of light to classical and quantum communication is an area of active research, because of the potential for increased energy efficiency and total data throughput [1–4]. One of the outstanding challenges to the practical application of spatial encoding is the slow rate at which distinct modes may be generated and detected. On the signal detection side, performing rapid and scalable spatial mode analysis is particularly desirable in the context of noisy and turbulent channels [5,6], since fast and complete channel characterization is necessary for adaptive-optic noise compensation in real time [7,8].

Several approaches to generating spatial modes have been previously demonstrated, but none attain an ideal combination of speed, efficiency, scalability, and reconfigurability. Actively and arbitrarily switchable spatial mode generation is currently performed with adaptive optics such as spatial light modulators (SLMs) or digital micromirror devices (DMDs) which alter the phase and amplitude front of the beam [9]. Both of these have refresh rates on the order of 1kHz, which limits potential data transmission rates. This limitation is especially problematic for quantum key distribution (QKD) [10] where the mode must be updated before every transmitted photon to ensure communication security. An alternative approach is to use multiple electro-optical modulators (EOMs) to illuminate one of several static phase plates, and recombine the resulting spatial modes using several beamsplitters [1]. This method has the advantage of being very rapid (on the order of tens of GHz), but the efficiency of such a device scales as 1/?, where ? is the number of desired modes. The mode conversion efficiency can be increased from ∼1/? to a fixed amount of loss by the use of multi-plane light conversion (MPLC) [11], where the desired spatial mode is generated through several reflections from a suitably patterned SLM. In order to produce different spatial modes at different times, MPLC would still require a number of switches that scales as ?. When ? becomes large, these approaches can become prohibitively expensive, limiting their scalability.

An acousto-optical modulator (AOM) is a common device that can act as a rapid, high-efficiency optical deflector with good multi-mode performance. AOMs are often used to control the propagation direction of light, similarly to how one would use a steerable mirror, acting like an N-port switch and allowing much greater efficiency and scaling capabilities than an ?-port beamsplitter array [12]. A double-pass AOM can be used to scan the frequency of a laser without affecting its spatial mode [13]. In this configuration, different RF frequencies applied to the AOM cause the output beam to be steered and re-focused on different regions of a retroreflection mirror, which can be seen as a folded 4-f optical system. AOMs are also used to synthesize arrays of rapidly movable focal spots for trapping atoms [14]. Most often the spatial structure of the beam remains Gaussian, just like the input beam, with dynamical control exerted only over the beam’s position or propagation direction.

Replacing the mirror of a folded 4-f system with a hologram (as shown in Fig. 1) allows different spatial modes to be encoded onto the reflected light when the input beam is steered to different regions of the hologram. By imprinting a linear phase grating onto the hologram, we can separate the light with the imprinted spatial mode structure from the zeroth order reflection. Thus, we can generate nearly-pure spatial modes of light with a rate limited only by the modulation rate of the AOM, which can readily reach ?=1MHz. When realizing the hologram using an SLM with a resolut