Optical core system

Abstract

An optical multiplication system comprises: a beam replication arrangement receiving input beams encoding an input vector and producing a plurality of spatially-separated replica beams; a modulator configured to encode values of a matrix and modulate the replica beams according to the encoded matrix values; and a light-summing optical arrangement converging the modulated replica beams to encode an output vector representing a result of multiplying the matrix by the input vector. The system comprises an optical input arrangement providing the input beams and comprising a three-dimensional light-guide providing the plurality of input beams to the beam replication arrangement as a two-dimensional array of input beams.

Description

[0001] OPTICAL CORE SYSTEM

[0002] The present invention relates to systems for optical computation of matrix-vector multiplications.

[0003] Optical processing is motivated by its ‘optical advantage’ over electronics.1Optical processors have a fundamentally larger bandwidth than electronic equivalents which allows for large wavelength-division multiplexing (parallel processing of multiple optical signals with subtly different wavelengths, on the same system) and lower computing latency. High spatial parallelism can be realised with optical signals using free-space components such as liquid crystal-based modulators or digital light processors where the spatial density of simultaneous encoding is superior. Optical signals experience approximately zero amplitude loss when transmitted through free space, and very low loss in fibre and waveguide, as well as through optical components with high-reflectivity or anti -reflectivity coatings. Optical computation dynamics are often approximately lossless when they involve propagation through some linear or nonlinear configuration (e.g., a lens for linear Fourier transform). Fanout (replication) of optical signals can be trivially performed with an array of lenses imaging the same object. Likewise, fan-in (summation) of optical signals can be trivially performed with a single lens focussing several incoming beams onto a detector.

[0004] Optical core processors (OCs) are potentially very advantageous as dedicated computation engines for large-scale matrix-vector operations which make up the majority of machine learning tasks. State-of-the-art artificial intelligence (Al) architectures such as large- scale machine learning (ML), deep learning (DL), and transformer-based large language models (LLMs) contain billions of trainable parameters, enabling their complex functionality. Using OCs for this role maximally utilises the optical advantage attributes listed previously and yields the largest performance improvement over incumbent technologies such as graphical and tensor processing units (GPUs and TPUs, respectively).

[0005] Analog optical computing hardware acceleration promises increased computing speed alongside improved energy and cost efficiency over the incumbent electronic processing units which support Al models. Optoelectronic hardware and optical computing architectures have the potential to decrease latency and energy consumption,2,3and the development of optical processor layers of appropriate size and complexity is accelerating. The recent realisation of a full optically trained optical neural network4validates the progress of the optical computing field and invites questions over which paths it should take to guarantee an impactful future.5 Many existing proposals for using optical processing for matrix-vector multiplication refer to systems on photonic platforms and are otherwise very slow, utilising 2D OLED screens. Some are technically optoelectronic processors which transform signals between optical and electronic to carry out different processes or use only passive free space optical components, which does not allow for easy or fast reconfigurability.

[0006] Several academic works have explored optical computing using free space optics (FSO). However, most of these are slow and not fit for large-scale, high-performance matrix-vector multiplier units. A concise overview of some key research is as follows.

[0007] Optical Neural Networks: Researchers at MIT & UCLA and elsewhere have developed all -optical neural networks using free-space diffractive optics. These systems can perform complex computations, image classification in this case, at the speed of light, achieving over 90% accuracy for handwritten digit recognition.6Miscuglio, et al. implements nonlinear activation functions using FSO whilst utilizing phase-change materials for tuneable nonlinearity.7They demonstrated the potential for high-speed, energy efficient neural computing.

[0008] Fourier Optics for Computing: Work has been done using Fourier optics principles in free space for fast matrix operations and convolutions, which are crucial for many computing tasks.8

[0009] Quantum Optical Computing: FSO has been used in experiments demonstrating quantum gates and quantum algorithms, leveraging the quantum properties of light.9

[0010] Reservoir Computing: Researchers have implemented optical reservoir computing systems using free-space optical elements for time-series prediction and pattern recognition from chaotic datasets.10

[0011] In conclusion, although research has explored the use of optical processors, the existing systems are not suited for commercial application for large-scale MVM operations. The present invention seeks to address some or all of the problems discussed above, as well as providing various other advantages as discussed in the following disclosure.

[0012] In accordance with a first aspect, there is provided an optical multiplication system comprising: an optical input arrangement configured to provide a plurality of input beams of light encoding an input vector; a beam replication arrangement configured to receive the plurality of input beams and produce a plurality of spatially-separated replica beams of each input beam; a modulator comprising a plurality of modulator elements configured to encode values of a matrix, wherein each modulator element is configured to modulate one of the replica beams received from the beam replication arrangement according to the respective encoded matrix value; and a light-summing optical arrangement configured to converge the modulated replica beams output from the modulator elements to encode an output vector representing a result of multiplying the matrix by the input vector, wherein the optical input arrangement comprises a three-dimensional light-guide providing the plurality of input beams to the beam replication arrangement as a two-dimensional array of input beams..

[0013] Using a three-dimensional light-guide to provide the plurality of input beams to the beam replication arrangement as a two-dimensional array of input beams means that the input beams can be provided in a very space-efficient and scalable configuration. The increased spatial density of input beams afforded by light guiding enhances the overall compute performance of the optical processor, by enabling an increase in the number of computations per clock cycle whilst retaining optical efficiency and electronic bandwidth versus a sparse arrangement of inputs. The variable arrangement of the light-guides can also be leveraged to create arbitrary input patterns which might offer computing advantage when paired with an optical encoder arrangement of related shape or size.

[0014] Optionally, the three-dimensional light guide comprises a two-dimensional array of elongate light-guiding elements. This allows individual light-guiding elements to be separately and securely connected to sources of light, maintaining a modular system which is compatible with various light sources and electronic drivers of light sources.

[0015] Optionally, the light-guiding elements comprise optical fibres, and the light guiding elements further comprise photonic waveguides connected to the optical fibres, the photonic waveguides configured to reduce a spacing of the light guiding elements in a plane perpendicular to a length of the light guiding elements. This allows the input beams to be more closely spaced than would be possible with optical fibres alone, reducing the size of the device and the entire arrangement of input beams.

[0016] Optionally, the optical input arrangement is configured to encode each element of the input vector using plural input beams having different intensities. This allows multiple bits of the input vector elements to be processed simultaneously, improving throughput and speed of processing.

[0017] Optionally, the intensities of the plural input beams corresponding to the same element of the input vector are related to one another by an exponential decay function. This allows the beams representing the multiple bits to be combined directly at the output to produce a result.

[0018] Optionally, the light-summing arrangement is configured to encode the output vector as a plurality of output beams encoding elements of the output vector. This allows multiple bits of the input vector elements to be processed simultaneously, improving throughput and speed of processing.

[0019] Optionally, the optical multiplication system further comprises an optical output arrangement configured to receive the plurality of output beams and produce an output corresponding to the output vector. This allows the system to directly produce an output for readout.

[0020] Optionally, the optical output arrangement comprises a three-dimensional light-guide configured to receive the plurality of output beams. Optionally, the three-dimensional light guide comprises a two-dimensional array of elongate light-guiding elements, optionally wherein the light-guiding elements comprise optical fibres, optionally large-mode area optical fibres. This allows the light to be guided to an appropriate detector in a modular fashion, with detectors and their accompanying electronics not being constrained in form-factor and positioning of the configuration of output beams.

[0021] According to a second aspect, there is provided a method of performing analogue computing, comprising: encoding an input vector using a plurality of input beams of light; producing a plurality of spatially-separated replica beams of each input beam; modulating the replica beams using a plurality of modulator elements encoding a matrix according to the respective encoded matrix value of each modulator element; and converging the modulated replica beams to encode an output vector representing a result of multiplying the matrix by the input vector, wherein the plurality of input beams is provided as a two-dimensional array of input beams.

[0022] Embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings in which corresponding reference symbols represent corresponding parts, and in which

[0023] Fig. 1 is a schematic illustrating the logical components of the optical multiplication system;

[0024] Fig. 2 is a flowchart of the method of optical multiplication carried out by the optical multiplication system; Fig. 3 illustrates the division of a ID input vector into a 2D array of elements;

[0025] Fig. 4 shows a 2D array of elongate light-guiding elements;

[0026] Fig. 5 shows a combination of optical fibres with photonic waveguides to form the array of elongate light-guiding elements;

[0027] Fig. 6 is a schematic of the optical multiplication system;

[0028] Fig. 7 shows the beam replication arrangement acting on a single input beam;

[0029] Fig. 8 shows the beam replication arrangement acting on plural input beams;

[0030] Fig. 9 shows an illustrative input and output of the beam replication arrangement;

[0031] Fig. 10 shows an exemplary array of input beams;

[0032] Fig. 11 shows how the replica beams may be produce from the array of Fig. 10 using a diffractive optical element;

[0033] Fig. 12 illustrates the loss of optical signal when replica beams are constructed from different diffractive orders of the input beams;

[0034] Fig. 13 shows the interaction of the replica beams with modulator elements;

[0035] Fig. 14 shows a reflective configuration of the modulator;

[0036] Fig. 15 shows a transmissive configuration of the modulator;

[0037] Fig. 16 shows the formation of the output beams using a microlens array; and

[0038] Fig. 17 illustrates a scheme for improving efficiency of beam replication.

[0039] The present disclosure provides an optical multiplication system for performing analogue computing, in particular matrix-vector multiplication operations.

[0040] The optical multiplication system provides a novel computation architecture designed for executing highly parallelised multiply-accumulate operations such as large-scale matrixvector multiplication (MVM) operations. The architecture operates with n-bit precision and throughput at frequencies higher than GHz, using a compact, all-optical, three-dimensional “optical core” (OC). The design and compact nature make the architecture modular and highly scalable for large-scale applications such as ML training and inference, unlike many of the existing approaches discussed above. The architecture uses a free-space optical approach, which means that the system’s footprint can be tailored as needed with the required imaging magnifications.

[0041] Existing OCs constructed with 3D optics in free space come in a variety of designs. The simplest designs use passive components to compute fixed processes for task-specific applications.6,11’12Configurations using reconfigurable free-space optical encoders might employ digital devices such as digital micromirror devices (DMDs) or liquid crystal-based devices such as liquid crystal spatial light modulators (LC-SLMs), to modulate matrix or vector elements at rates from kHz to MHz.7,13 15Some of these are restricted to small matrix sizes (i.e. « 1000 x 1000) or use non-established materials for optical modulation.16,17Recent academic or technical works are often focussed on utilising the direct optical input capabilities of optical MVM, for example processing live images directly from source or from an intermediary screen. However, these do not prioritise throughput and compatibility with data centre hardware and software architectures.3These are primarily experimental laboratory setups which lack scalability and the ability to directly integrate with other systems.

[0042] Previous publications have used spatial light modulators (SLMs) as free space optical encoders for MVM and utilised this approach in demonstrations of both hybrid-trained and optically-trained optical neural networks.4,18,19Strategies for correcting the performance of optical encoders (in particular, SLMs) have also been investigated.20However, there have been no examples in the academic literature of an optical matrix-vector multiplier designed for throughput at frequencies higher than GHz and employing 2D fan-in and fan-out for massively parallel processing using a free-space configuration.

[0043] The present analogue OC can perform MVM with a computational capacity of over 1,000 tera operations per second (TOPS) (for example up to 10,000 TOPS or more) with high energy efficiency. The three-dimensional optical configuration of the OC provides scalability superior to that of two-dimensional electronic or integrated photonic processors, and improved optical aperture over one-dimensional equivalents. The use of free-space optical components offers improved stability and fabrication simplicity compared to fibre or integrated processors. Additionally, the architecture is highly adaptable to various other architectural variants with respect to the optical components, methodologies, and light display equipment. The three-dimensional configuration, with densely packed input and output arrays and primarily passive components, also promises a quadratic scaling of advantage in cost, energy efficiency, and computational intensity over incumbent electronics which could unlock huge improvements on the impact of LLMs and DL models.

[0044] The fundamental processes required for MVM are: (1) the element-wise multiplication of two arrays, and (2) the row / column-wise summation of those resulting products.

[0045] Fig. 1 is a schematic that illustrates the logical components of the optical multiplication system 1 and the flow of light through the optical multiplication system 1. Examples of more specific components that can be used to implement each logical component are also given in Fig. 1.

[0046] Fig. 2 is a flowchart that shows how these basic steps are split up into operations that are performed by the optical multiplication system 1 to carry out a method of performing analogue computing. Together, these illustrate how parallel MVM is implemented by the OC, as will be discussed in more detail below.

[0047] The optical multiplication system 1 comprises an optical input arrangement 10, an optical core processor 20, and may comprise an optical output arrangement 30. The optical input arrangement 10 produces input beams of light encoding the input vector. The optical core processor 20 receives the input beams and performs the MVM with a matrix encoded in the optical core processor 20. Finally, the optical output arrangement 30 receives the optical output of the optical core processor 20 and produces an output corresponding to the result of the MVM that can be transmitted to other systems, for example an electrical signal.

[0048] The components of the optical multiplication system 1 are coupled together through free-space transmission of light between the different components. The ability to manipulate light in free space facilitates modularity and dynamic reconfigurability, making it ideal for adaptive computing paradigms. As such, the present optical core processor 20 is more scalable than photonic processors which are fabricated on 2D substrates and therefore a constrained 2D topology. For example, Mach-Zender interferometer meshes receive X input signals and require X-l layers of interferometer units to perform full MVM.

[0049] The optical multiplication system 1 may comprise an optical input arrangement 10 configured to provide a plurality of input beams of light encoding S 10 an input vector.

[0050] The optical input arrangement 10 may comprise an emitter array 11, as shown in Fig. 1. The emitter array 11 may be implemented as a plurality of modulated optical emitters configured to produce the plurality of input beams. Any suitable emitters could be used, such as laser diodes (e.g. a vertical -cavity surface-emitting laser, VCSEL). The emitters may be binary emitters.

[0051] Each emitter may provide one of the input beams, such that the number of emitters is equal to the number of input beams. Alternatively, the emitter array 11 could be implemented using one or more light sources and a plurality of optical modulators (such as pixels of an LCD screen) configured to modulate light from the one or more light sources to produce the plurality of input beams. In this case, the number of light sources may be smaller than the number of input beams, but the number of modulators may be equal to the number of input beams. Any suitable modulator could be used, such as an absorptive modulator or a refractive modulator. Each vector is prepared by simultaneously writing each vector element to one or more emitters and / or modulators. This is carried out by control electronics (not pictured).

[0052] The encoding of the input vector into the plurality of input beams can be performed in a variety of different ways depending on the priorities of the specific application. For example, the optical input arrangement 10 may be configured to encode the input vector using temporal modulation of the input beams. The emitters and / or modulators configured to produce the input beams are preferably able to change their output at very high frequencies, in order to maximise the throughput of the system 1. For example, the emitters and / or modulators may have a modulation frequency of at least 1 MHz, optionally at least 100MHZ, optionally at least 1GHz, optionally at least 5GHz, optionally at least 10GHz, optionally at least 20GHz, optionally at least 40GHz.

[0053] The number of input beams used to encode each element of the input vector may also vary. For example, a single input beam may be used for each element of the input vector. Binary emitters and / or modulators could be used to deconstruct each n-bit analogue vector element as a bitstream of n binary signals in the corresponding input beam. In this case, each element would be transmitted by a single beam over n timesteps of the emitters and / or modulators. Alternatively, a multi-level emitter and / or modulator with more than two levels of output could be used to encode the elements.

[0054] Alternatively or additionally, the optical input arrangement 10 may be configured to encode each element of the input vector using plural input beams. For example, for an n-bit vector element, n input beams may be used to encode each bit of the input beam using a separate input beam. This maximises the speed of processing of each individual vector element, but reduces bandwidth.

[0055] A combination of the two approaches may also be used, with an n-bit element being encoded using fewer than n input beams, but with temporal modulation of each individual input beam to encode plural bit of the input beam with each corresponding input beam.

[0056] The plural input beams corresponding to the same element of the input vector may have different intensities. For example, the intensities of the plural input beams corresponding to the same element of the input vector may be related to one another by an exponential decay function. This can allow the input beams corresponding to the same element to be directly summed up to provide an output intensity after processing by the OC. For example, a four-bit element could be represented by four beams having relative intensities of 1, 0.5, 0.25, 0.125. The direct superposition of the four beams would then provide an intensity proportional to the value of the element. This scheme is described in detail in PCT / GB2024 / 051866, published as WO 2025 / 017301 Al, the contents of which are incorporated herein by reference.

[0057] As well as the encoding in terms of the number and temporal variation of the input beams, the optical input arrangement 10 also takes the ID input vector and breaks and folds it into a J*K 2D arrangement, as shown in Fig. 3. This allows a more compact arrangement of the system especially for longer vectors.

[0058] To facilitate this, the optical input arrangement 10 comprises a three-dimensional lightguide 12 providing the plurality of input beams to the beam replication arrangement 20 as a two-dimensional array of input beams. The three-dimensional light-guide 12 can provide the plurality of input beams in an arbitrary arrangement and with arbitrary transverse mode profiles depending on what is most suitable for a particular application.

[0059] The three-dimensional light guide 12 may comprise a two-dimensional array of elongate light-guiding elements, for example optical fibres 14, as shown in Fig. 4. Any suitable type of optical fibre can be used, such as single-mode fibres, multi-mode fibres, or multi-core fibres.

[0060] The two-dimensional array of elongate light-guiding elements preferably has a small pitch between the light-guiding elements in order to maximise the compactness of the light guide 12 and the system 1 as a whole. For example, the pitch between light-guiding elements may be at most 1mm, optionally at most 500pm, optionally at most 250pm, optionally at most 130pm, 100pm.

[0061] The array of elongate light-guiding elements may be a square array as shown in Fig. 4. This allows the light-guiding elements to be provided in a high density arrangement, for example, 100 light-guiding elements per mm2. Alternatively, the array of elongate lightguiding elements may be a hexagonal array. A hexagonal array can provide for tighter packing of light-guiding elements such as the optical fibres 14 that will generally have a circular cross-section. This can allow a higher density arrangement of the light-guiding elements, for example, 115 light-guiding elements per mm2.

[0062] As shown in Fig. 5, the light guiding elements may further comprise photonic waveguides 16 connected to the optical fibres 14. The photonic waveguides 16 may be configured to reduce a spacing of the light guiding elements in a plane perpendicular to a length of the light guiding elements. This is illustrated in Fig. 5 for a single layer of the light guide 12 (which would be replicated in the direction perpendicular to the page to produce the two-dimensional array). Photonic waveguides can be manufactured with a smaller spacing than is possible with an array of optical fibres alone, due to the minimum effective diameter with which it is possible to manufacture suitable optical fibres, which is typically around 250pm. The pitch between photonic waveguides may be at most 100pm, optionally at most 50pm, optionally at most 30pm, optionally at most 20pm. This has advantages particularly when a DOE is used for the beam replication arrangement, as will be described further below.

[0063] The use of 2D arrays of light-guiding elements offers multiple advantages, such as:

[0064] 1. Enhanced data throughput with high-speed parallel transmission on more matrix elements per mm2, for higher spatial compute density.

[0065] 2. Improved energy efficiency through optimized optical coupling (optical losses are reduced due to the lower optical extent [etendue] of the dense input array), and optical fan-out (or beam replication, discussed further below).

[0066] 3. Increased system robustness via redundancy and distributed architecture.

[0067] 4. Modular design enabling flexible scaling and easy maintenance.

[0068] 5. Compatibility with a wide range of photonic components for versatile applications.

[0069] The optical input arrangement 10, and in particular the light-guiding elements, may be further designed to control the nature and combination of optical transverse modes supported within and launched from the three-dimensional light-guide of the optical input arrangement 10 into the free-space optical processor. This enables control of the beam parameter product (i.e., minimisation of BPP by selectively supporting single transverse mode propagation), and illumination profile on the optical encoder (i.e., supporting the illumination of individual or multiple encoder pixels).

[0070] For example, each input beam may be provided as a single transverse mode, optionally a TEMoo Gaussian mode. This can help to provide an optimal combination of spot size and divergence / numerical aperture, i.e., a minimum possible beam parameter product (BPP). Alternatively each input beam may comprise a plurality of transverse modes. Multimode input beams allow for modal multiplexing or formation of different illumination profiles on the modulator elements of the modulator 24.

[0071] Once the input beams have been arranged into appropriate configuration by the optical input arrangement 10, the beams are transmitted to the optical core processor (OC) 20. Fig. 6 shows a design overview for an exemplary implementation of the OC 20. The OC 20 receives the input beams and combines the input beams in an appropriate way to perform the desired optical processing operation, such as an MVM.

[0072] The OC 20 comprises three main components: a beam replication arrangement 22, a modulator 24, and a light-summing optical arrangement 26. These will be discussed in turn below.

[0073] The beam replication arrangement 22 is configured to receive the plurality of input beams and produce S20 a plurality of spatially-separated replica beams of each input beam. This process is also referred to as optical “fan-out”.

[0074] The beam replication arrangement 22 is configured to produce the plurality of spatially- separated replica beams of each input beam as a two-dimensional array of replica beams. The replica beams may be evenly spaced in both dimensions of the 2D array. Each input beam may be replicated into at least 4 replica beams, optionally at least 16, optionally at least 32, optionally at least 64, optionally at least 128, optionally at least 256, optionally at least 1024, optionally at least 4096.

[0075] The beam replication arrangement 22 is couple through free space to the optical input arrangement 10, such that the input beams are transmitted to the beam replication arrangement through free space. The fan-out process also occurs in free space, such that the replica beams are transmitted from the beam replication arrangement 22 to the modulator 24 through free space.

[0076] This is a key advantage of the present system compared to photonic systems. There is limited scope on photonic processor systems for optical fan-out, which is able to massively enhance the parallelism of the present system in a highly scalable way. 3D optical configurations such as the present invention benefit from quadratic scaling with number of input signals X. Because of the optical fan-out capabilities, each input is not only replicated but is replicated in two dimensions in a 2D array, increasing the number of active elements on the optical encoder to X2. This is also an inherent advantage of the present system over incumbent electronic systems whose processor units are again fabricated in 2D and can only be accessed across the 2D substrate. Moreover, 2D photonic meshes must be interspersed with electronic drivers and electrodes increasing the device footprint to an impractical extent. In addition, waveguide photonic approaches create cascading optical loss, whereas the use of free space transmission between components in the present system minimises optical loss. The fan-out process is illustrated in more detail in Fig. 7 to Fig. 9. The input beams are provided as a 2D (JXK) array, as discussed above. The input beams are transmitted from the optical input arrangement 10 through free space to the beam replication arrangement 22.

[0077] As illustrated in Fig. 7 for a single input beam, the beam replication arrangement 22 generates PXQ replica beams (in this case, 10x10) of each input beam, arranged in a 2D array. The replica beams are produced by passively replicating the incoming signal channels.

[0078] As shown in Fig. 8, when this is repeated for all JxK input beams (in this case, a 3x3 2D input), this results in an MxN array of replica beams (where M = JP and N = KQ) which in total contain the same amount of power as the sum of the initial JXK array of input beams. Each replica beam will thus contain a factor PQ times less power than the input beam it originates from. Each region circled on the right-hand side of Fig. 8 is a replica of the 2D input that is tiled across the image plane. Each of the PxQ sections then contains a replica of the original JxK input. The DOE 28 may arrange the replica beams such that the pitch of the individual optical signals is reduced but the spot size is preserved.

[0079] The overall effect is shown again in Fig. 9. The gradation in the colour contrast denotes the correspondence between the original input beams and the replica beams, and does not necessarily indicate that the input beams and / or replica beams have different intensities (although in some embodiments they may, as discussed above).

[0080] As shown in the exemplary arrangement of the figures, the beam replication arrangement 22 may comprise a diffractive optical element 28. The DOE 28 may implement a bespoke diffractive phase mask to produce the required pattern of replica beams. Using a diffractive optical element allows the beam replication arrangement to create complex patterns of replica beams using a very compact arrangement of optical components. However, this is not essential, and in other embodiments, other optical components may be used to replicate the input beams. For example, the beam replication arrangement 22 may comprise a pair of micro-lens arrays arranged in series with a focusing lens, a similar arrangement as is sometimes used for an imaging homogenizer.

[0081] If the intensity variation of the input beams described above is implemented in the beam replication arrangement 22, a separate DOE could be used for the intensity variation, optionally in combination with a DMD. Alternatively, where a DOE is used to produce the replica beams, the intensity variation could be implemented by the same DOE as is used for producing the replica beams in a single operation. Where a DOE 28 is used, each replicated output illustrated in Fig. 9 may be generated by overlapping different diffractive orders of different input beams. This is illustrated in Fig. 10 and Fig. 11. In Fig. 10, three of the input beams are labelled as beams A, B, and C for illustrative purposes.

[0082] In Fig. 11, the top row of the corresponding replicated output is shown. Due to factors such as the diameter of the optical fibres 14 in the input arrangement, it may not be possible to bring the input beams A, B, and C close enough together to match the desired spacings between corresponding replica beams in a single group 23 of replica beams. For example, the pitch of the modulator elements on the modulator 24 may be smaller than the minimum achievable spacing of the input beams.

[0083] Using a two-dimensional array of elongate light-guiding elements as part of the three- dimensional light-guide of the optical input arrangement can help to solve this problem by reducing the spacing of the input beams, as described above.

[0084] Alternatively or additionally, to solve this problem, each group 23 of replica beams can be formed by overlapping the replica beams generated by different diffractive orders of the corresponding input beam. For example, the top row of group 23 is formed by overlapping the zeroth order of input beam C, the first order of input beam B, and the second order of input beam A.

[0085] A disadvantage of this approach is illustrated in Fig. 12. Overlapping different diffractive orders means that there is a large number of incomplete groups of replica beams generated that cannot be used. Only the complete replicas in the highlighted inner box can be used in the MVM processing. This results in a lot of wasted light, leading to reduced efficiency and increased power requirements.

[0086] A partial solution to the issue of wasted optical power is to increase the effective matrix size by processing the MVM results of incomplete groups of replica beams and summing them to form complete groups of replica beams, yielding additional MVM results, as illustrated in Fig. 17. A variation of this approach would be to reduce the optical fan-out dimension while maintaining the same effective matrix size by processing and summing incomplete groups of replica beams. These strategies increase the optical efficiency of the system. Signals from the incomplete groups may be summed in the optical domain before detection, or in the electronic domain after detection at the photodiode. The electronic summation might involve the summing of the resulting photocurrents generated at each photodetector.

[0087] To optimise the optical efficiency of the system, use of a combination of optical fibres 14 and photonic waveguides 16 in the input arrangement 10, as discussed above, is advantageous. The use of photonic waveguides 16 can reduce the spacing of the input beams to match the pitch of the modulator elements. Therefore, each complete group of replica beams can be formed from the same diffractive order of the input beams. This reduces wasted light, because substantially all the generated diffractive order replicas of each input can be used.

[0088] The beam replication arrangement 22 may comprise further optical components to properly condition and position the input beams for replication. As shown in Fig. 7 and Fig. 8, the beam replication arrangement 22 may use a configuration of two-dimensional lens arrays and a 4f imaging system. The DOE 28 is then placed at the Fourier plane of the 4f imaging system. This design enables flexibility by allowing the OC 20 to be decoupled from the input arrangement, because the operation of the beam replication arrangement 22 can be made more independent of the exact arrangement of the input beams. This can enable functions such as wavelength multiplexing or interoperability with different types of emitter.

[0089] The OC 20 further comprises a modulator 24, also referred to as an optical encoder. An exemplary modulator 24 is illustrated schematically in Fig. 13. The modulator 24 comprises a plurality of modulator elements configured to encode values of a matrix. The plurality of modulator elements is preferably arranged in rows and columns to form a 2D array of modulator elements. The replica beams from the beam replication arrangement 22 are transmitted through free space to the modulator, where the replica beams are imaged on to individual pixels of the modulator. Thereby, the modulator 24 modulates S30 the replica beams according to the respective encoded matrix value of each modulator element.

[0090] The pitch of the replica beams in each dimension is chosen to match the pitch of the rows and columns of the modulator 24, such that each modulator element receives exactly one replica beam. Each modulator element is configured to modulate one of the replica beams received from the beam replication arrangement 22 according to the respective encoded matrix value.

[0091] As discussed above, the beam replication arrangement 22 may be configured to recover the individual spot sizes originating from the optical input arrangement 10, such that each replica beam has a spot size matching the spot size of the corresponding input beam. This may be achieved using a 4f imaging system. By controlling the magnification of the replica beams in the beam replication arrangement 22, the spot size of the replica beams can be controlled to ensure each modulator element of the modulator 24 is filled as efficiently as possible by the replica beam. This principle is scalable to any arbitrary dimensions of input beam and / or replica beam array and modulator 24.

[0092] As shown in Fig. 13, the modulator 24 applies weights to the replica beams according to the encoded matrix values. In particular, each modulator element applies a weight to the corresponding replica beam according to a particular encoded element of the matrix.

[0093] The modulator 24 can be considered to be logically divided into P*Q regions, each of which receives a replica of the entire input beam array (representing the input vector) and represents a section of the matrix. Each of the PxQ regions contains plural modulator elements (in particular, JxK modulator elements) such that each replica beam (representing an element of the input vector) is incident on a single modulator element (also referred to as a pixel) of the optical encoder. Thereby, each element of the input vector is multiplied with the respective element of the matrix, with all of the multiplications occurring simultaneously in parallel.

[0094] The input beams (and thereby the replica beams) are modulated at very high rates (e.g. optionally 1GHz, optionally 10GHz, optionally 100GHz or higher) such that the throughput of parallel computations is extremely high. Matrix values are fed to the modulator 24 via appropriate encoder driver circuitry. The modulator elements may be modulated at a much lower rate than the input beams, for example a factor of 103- 106times lower. The modulator elements may be modulated at a frequency of at least 1kHz, optionally at least 10kHz, optionally at least IMhz.

[0095] Programming of the encoder driver enables desired sub-sections of the modulator 24 to be updated separately and more quickly than a full matrix update. Depending on the choice of modulator 24 and its programming, the precision of the matrix encoding can be varied to n-bit precision. Depending on the exact nature of the matrix operations that are to be computed, this can enable improved efficiency by appropriate design of how the matrix values are transmitted to the modulator 24.

[0096] Any suitable modulator 24 may be used. For example, the modulator 24 may comprise a spatial light modulator, SLM, (such as a liquid-crystal spatial light modulator, LC-SLM, or liquid-crystal on silicon spatial light modulator, LCoS-SLM), a digital micromirror device, DMD, or an optical micro-electromechanical, MEM, system. Which device is used depends on the priorities of a specific use case. For example, a DMD might be preferable for a 1 -bit MVM where speed is a priority, because DMDs can be modulated very quickly. An LCoS SLM might be preferable for an 8-bit MVM, where system precision is more important than processing speed. Combinations of different devices may also be used. For example, a DMD coupled with an LCoS SLM for intensity correction and n-bit precision can balance speed and precision. A single multi quantum well SLM may provide very high speed, n-bit precision.

[0097] The modulator 24 may be arranged in a transmissive or reflective configuration, dictating the relative positioning in space of the optical fan-out and fan-in components. Fig. 14 and Fig. 15 illustrate examples of these two options.

[0098] Fig. 14 shows an implementation using a reflective LCoS-SLM as the modulator 24. Although LCoS-SLMs are polarisation-sensitive, this configuration is designed to be quasipolarisation-insensitive. A polarizing beam splitter (PBS) splits the incoming beam into s- and p- polarised light. The half-wave plate (X / 2) rotates the two components so that they hit the SLM for amplitude modulation at 45 degrees to the polarisation-sensitive axis of the SLM. The SLM is divided into two sections, both encoding the same matrix, to modulate the s- and p- polarised components according to the same matrix elements. After amplitude modulation by the SLM, both s- and p- polarised light merge at the PBS, after yet again being rotated by 45 degrees by a second pass through the half-wave plate.

[0099] Fig. 15 shows the same scheme implemented using a transmissive LCoS-SLM. In this implementation, two separate PBSs and half-wave plates are required, but the principles are otherwise the same as for the reflective arrangement in Fig. 15.

[0100] Moreover, a combination of matrix encoders can be used to enable maximum update rate. The modulator 24 may comprise a first component configured to change a modulation applied to the replica beams on a first timescale, and a second component configured to change a modulation applied to the replica beams on a second timescale that is longer than the first timescale. Optionally, the modulation applied to the replica beams by the second component is static. The second component allows the modulator 24 to selectively attenuate the replicated beams to unify their intensities or selectively weight them before dynamic calculations are carried out.

[0101] For example, the first component may comprise a rapidly modulating DMD device and the second component may comprise a spatial light modulator such as an LCoS SLM displaying a static pattern or a static intensity correction mask, thus providing an attenuation mask which applies the desired intensity correction to the input beams or weight elements. This methodology can also be adopted to increase the effective bit depth of the matrix deployed, by weighting different elements using the desired attenuation mask.

[0102] The OC 20 further comprises a light-summing optical arrangement 26 configured to converge S40 the modulated replica beams output from the modulator elements to encode an output vector representing a result of multiplying the matrix by the input vector. This process is also referred to as optical “fan-in”. As shown in Fig. 6, optical fan-in carries out local summation of the P*Q regions of the modulator 24. The resulting sums can then be coupled to detectors in the optical output arrangement 30.

[0103] The light-summing arrangement 26 may comprise an optical fan-in component such as a microlens array (MLA) 32. Modulated replica beams from each modulator region, representing products of input vector elements and matrix elements, are incident on a designated region of the optical fan-in component, where they experience local summation. Thereby, each array of product signals is condensed to a separate location in space where the sum of the overlapping signals can then be received by the optical output arrangement 30.

[0104] For example in Fig. 16, each microlens of the MLA focuses the modulated replica beams from one region of the modulator 24 into a localised spot to produce an output beam. Thereby, the light-summing arrangement 26 is configured to encode the output vector as a plurality of output beams encoding elements of the output vector.

[0105] The spatial size of the output beams produced from each modulator region is dictated by the design of the fan-in component. It is preferably equal to or less than the aperture size of the corresponding receiving component (for example an optical fibre core or detector element) of the output arrangement 30 on which they are incident. This ensures the fidelity of the summation and thus the accuracy of the MVM. Additionally, the angle of incidence onto the receiving component of the output arrangement 30 of every output beam must be equal to or less than the angle of acceptance of the receiving component.

[0106] Similar to the beam replication arrangement 22, the light-summing optical arrangement 26 may comprise further optical components to properly condition and position the modulated replica beams for summation by the optical fan-in component. For example, a 4f imaging system may be used between the output of the modulator 24 and the optical fan-in component. The optical fan-in component is placed a distance f from the imaging plane of the 4f system, where f in this case is the local length of the optical fan-in component, such that the beams are collimated after passing through the optical fan-in component. As for the beam replication arrangement, this design enables flexibility by allowing the OC 20 to be decoupled from the output arrangement 30.

[0107] Following the OC 20, the optical multiplication system 1 may further comprise an optical output arrangement 30 configured to receive the plurality of output beams and produce S50 an output corresponding to the output vector based on the converged beams.

[0108] The optical output arrangement 30 may comprise a three-dimensional light-guide configured to receive the plurality of output beams. The light-guide of the output arrangement may be similar to the light-guide of the input arrangement. For example, the three-dimensional light guide may comprise a two-dimensional array of elongate lightguiding elements 34, such as shown in Fig. 6. The light-guiding elements may be optical fibres. The properties of the output fibre array including fibre core and cladding diameters and materials, are customisable to suit end-user needs. In one implementation, large mode area (LMA) optical fibres might be used to maximise the coupling efficiency of summed optical signals from free space to the fibre array.

[0109] The optical output arrangement 30 receives the P*Q summed signal groups from the OC 20. For example, each summed signal group may be focussed into the core of one of the optical fibres of the light guide.

[0110] The optical output arrangement 30 may further comprise a detector arrangement configured to individually read out each of the plurality of output beams. The output beams may be directed to the detector arrangement by the light guide of the output arrangement, such that the light guide couples the detector arrangement to the output of the OC 20.

[0111] The detector arrangement may comprise an array of optical detectors, for example an array of photodiodes. Within the detector arrangement, the summed optical signals represented by the output beams from the OC 20 are converted to electronic signals which are then transferred to receiver electronics for processing. The result of the MVM calculation is read out of the receiver electronics at the same clock rate as the input signals are modulated.

[0112] As mentioned above, elements of the input vector may be encoded with plural input beams, and the intensities of the input beams may be related to one another by an exponential decay function. A similar effect can also be achieved when an n-bit analogue vector element is encoded as a bitstream of n binary signals in a single input beam. In this latter case, the detector arrangement may be tuned to have an appropriate RC time constant (matching the clock rate of the modulation of the input beam used to encode the bitstream) to exponentially weight the MVM output data to reconstruct the n-bit analogue result. For example, a transimpedance amplifier could be used with an appropriate time constant, such that in the sensing window of the amplifier the first bit of the bitstream corresponding to a single input vector element is amplified with a relative weight of 1, the second bit with a relative weight of 0.5, the third with a relative weight of 0.25, and so on. In this way, the overall value of the vector element represented by the entire bitstream can be accumulated in a single value quickly and straightforwardly without the need for further post-processing.

[0113] The present system may further comprise an electronic control board to receive input data from a host machine and transmit output data back to the host system. The input data could comprise the input vectors and matrix that are to be multiplied. Based on the input data, the electronic control board can control the input arrangement 10 and the modulator 24 to perform the MVM.

[0114] This makes the system 1 highly modular and suitable for integration in a wide range of existing electronic systems. Different host system interfaces could be implemented by changing the electronic control board while the rest of the system 1 remains unchanged. The electronic control board may be further configured to apply some post-processing steps to the resulting data, to decrease the computational burden on the host processor.

[0115] The system 1 can be implemented as a plugin card, for example in the style of a PCIe plugin card. This provides a small, compact, implementation suitable for desktop and workstation applications. Larger implementations of the system 1 with more input beams and modulator elements would provide increased throughput and scale more suitable for datacentres, for example implemented in a unit matching standard server rack dimension.

[0116] The OC 20 may also be configured to perform basic logic and mathematical functions, such as one or more of addition, subtraction, matrix transform, convolution, differentiation, and rotation. The availability of these operations removes the need for data that is being processed on the MVM system to be copied to the host system for an operation and then back to the MVM for continued processing. This removes memory copy overhead associated with moving data to and from the host system.

[0117] The use of three-dimensional light guides, such as optical fibre arrays, to connect the OC 20 to the input and output arrangements provide significant flexibility of system architecture. Moreover, the layout of the OC 20 can be fully customised and appropriately condensed by choice of beam-steering optics. For example, the OC 20 might be laid out in a linear configuration with a transmissive optical encoder, or in a raster configuration to consume least volume for compatibility with existing server racks in data centres.

[0118] The following numbered clauses define further optional configurations of the present disclosure. These are not the claims of the present application, which follow under the heading “CLAIMS”.

[0119] 1. An optical multiplication system comprising: an optical input arrangement configured to provide a plurality of input beams of light encoding an input vector; a beam replication arrangement configured to receive the plurality of input beams and produce a plurality of spatially-separated replica beams of each input beam; a modulator comprising a plurality of modulator elements configured to encode values of a matrix, wherein each modulator element is configured to modulate one of the replica beams received from the beam replication arrangement according to the respective encoded matrix value; and a light-summing optical arrangement configured to converge the modulated replica beams output from the modulator elements to encode an output vector representing a result of multiplying the matrix by the input vector, wherein the optical input arrangement comprises a three-dimensional light-guide providing the plurality of input beams to the beam replication arrangement as a two- dimensional array of input beams.

[0120] 2. The system of clause 1, wherein the beam replication arrangement comprises a diffractive optical element.

[0121] 3. The system of clause 1, wherein the beam replication arrangement comprises a pair of micro-lens arrays arranged in series with a focusing lens.

[0122] 4. An optical multiplication system comprising: a beam replication arrangement configured to receive a plurality of input beams of light encoding an input vector and produce a plurality of spatially-separated replica beams of each input beam; a modulator comprising rows and columns of modulator elements configured to encode values of a matrix, wherein each modulator element is configured to modulate one of the replica beams received from the beam replication arrangement according to the respective encoded matrix value; and a lightsumming optical arrangement configured to converge light output from the modulator elements to encode an output vector representing a result of multiplying the matrix by the input vector, wherein the beam replication arrangement comprises a diffractive optical element.

[0123] 5. The system of clause 4, further comprising an optical input arrangement configured to provide a plurality of input beams of light encoding an input vector;

[0124] 6. The system of clause 5, wherein the optical input arrangement comprises a three- dimensional light-guide providing the plurality of input beams to the beam replication arrangement as a two-dimensional array of input beams.

[0125] 7. The system of any of clauses 1 to 3 or 6, wherein the three-dimensional light guide comprises a two-dimensional array of elongate light-guiding elements.

[0126] 8. The system of clause 7, wherein the light-guiding elements comprise optical fibres.

[0127] 9. The system of clause 7 or 8, wherein the light guiding elements comprise photonic waveguides.

[0128] 10. The system of clause 8, wherein the light guiding elements further comprise photonic waveguides connected to the optical fibres, the photonic waveguides configured to reduce a spacing of the light guiding elements in a plane perpendicular to a length of the light guiding elements.

[0129] 11. The system of any preceding clause, wherein the optical input arrangement comprises: i) a plurality of modulated optical emitters configured to produce the plurality of input beams; or ii) one or more light sources and a plurality of optical modulators configured to modulate light from the one or more light sources to produce the plurality of input beams.

[0130] 12. The system of any preceding clause, wherein the optical input arrangement is configured to encode the input vector using temporal modulation of the input beams.

[0131] 13. The system of any preceding clause, wherein the optical input arrangement is configured to encode each element of the input vector using plural input beams.

[0132] 14. The system of clause 13, wherein the plural input beams corresponding to the same element of the input vector have different intensities.

[0133] 15. The system of clause 14, wherein the intensities of the plural input beams corresponding to the same element of the input vector are related to one another by an exponential decay function.

[0134] 16. The system of any preceding clause, wherein the beam replication arrangement is configured to produce the plurality of spatially-separated replica beams of each input beam as a two-dimensional array of replica beams.

[0135] 17. The system of any preceding clause, wherein the modulator comprises a spatial light modulator, for example a liquid-crystal spatial light modulator.

[0136] 18. The system of any preceding clause, wherein the modulator comprises a digital micromirror device.

[0137] 19. The system of any preceding clause, wherein the light-summing arrangement comprises a microlens array, and optionally further comprises a spherical lens.

[0138] 20. The system of any preceding clause, wherein the light-summing arrangement is configured to encode the output vector as a plurality of output beams encoding elements of the output vector.

[0139] 21. The system of clause 20, wherein the optical multiplication system further comprises an optical output arrangement configured to receive the plurality of output beams and produce an output corresponding to the output vector based on the plurality of output beams.

[0140] 22. The system of clause 21, wherein the optical output arrangement comprises a three- dimensional light-guide configured to receive the plurality of output beams.

[0141] 23. The system of clause 22, wherein the three-dimensional light guide comprises a two- dimensional array of elongate light-guiding elements, optionally wherein the light-guiding elements comprise optical fibres, optionally large-mode area optical fibres.

[0142] 24. The system of any of clauses 21 to 23, wherein the optical output arrangement comprises a detector arrangement configured to individually read out each of the plurality of output beams.

[0143] 25. A method of performing analogue computing, comprising: encoding an input vector using a plurality of input beams of light; producing a plurality of spatially-separated replica beams of each input beam; modulating the replica beams using a plurality of modulator elements encoding a matrix according to the respective encoded matrix value of each modulator element; and converging the modulated replica beams to encode an output vector representing a result of multiplying the matrix by the input vector, wherein one or both of: i) the plurality of input beams is provided as a two-dimensional array of input beams; and ii) the replica beams are produced using a diffractive optical element.

[0144] 26. The method of clause 25, further comprising producing an output corresponding to the output vector based on the converged beams. References

[0145] 1. McMahon, P. L. The physics of optical computing. Preprint at https: / / doi.org / 10.48550 / arXiv.2308.00088 (2023).

[0146] 2. Guo, X., Barrett, T. D., Wang, Z. M. & Lvovsky, A. I. B ackpropagation through nonlinear units for the all-optical training of neural networks. Photon. Res., PRJ 9, B71-B80 (2021).

[0147] 3. Wang, T. et al. An optical neural network using less than 1 photon per multiplication. Nat Commun 13, 123 (2022).

[0148] 4. Spall, J., Guo, X. & Lvovsky, A. I. Training neural networks with end-to-end optical backpropagation. Preprint at https: / / doi.org / 10.48550 / arXiv.2308.05226 (2023).

[0149] 5. Matuszewski, M., Prystupiuk, A. & Opala, A. The role of all-optical neural networks. Preprint at http: / / arxiv.org / abs / 2306.06632 (2023).

[0150] 6. Lin, X. et al. All-optical machine learning using diffractive deep neural networks. Science 361, 1004-1008 (2018).

[0151] 7. Miscuglio, M. et al. Massively parallel amplitude-only Fourier neural network. Optica, OPTICA 7, 1812-1819 (2020).

[0152] 8. Psaltis, D. & Farhat, N. Optical information processing based on an associative- memory model of neural nets with thresholding and feedback. Opt. Lett., OL 10, 98-100 (1985).

[0153] 9. Martin-Lopez, E. et al. Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nature Photon 6, 773-776 (2012).

[0154] 10. Rafayelyan, M., Dong, J., Tan, Y, Krzakala, F. & Gigan, S. Large-Scale Optical Reservoir Computing for Spatiotemporal Chaotic Systems Prediction. Phys. Rev. X 10, 041037 (2020).

[0155] 11. Chang, J., Sitzmann, V., Dun, X., Heidrich, W. & Wetzstein, G. Hybrid optical- electronic convolutional neural networks with optimized diffractive optics for image classification. Sci Rep 8, 12324 (2018).

[0156] 12. Goodman, J. W, Dias, A. R. & Woody, L. M. Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms. Opt. Lett., OL 2, 1-3 (1978).

[0157] 13. Wang, T. et al. Image sensing with multilayer nonlinear optical neural networks. Nat. Photon. 17, 408-415 (2023).

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[0159] 15. Zhou, T. et al. Large-scale neurom orphic optoelectronic computing with a reconfigurable diffractive processing unit. Nat. Photonics 15, 367-373 (2021). 16. Tang, Y. et al. Device-system Co-design of Photonic Neuromorphic Processor using Reinforcement Learning. Preprint at http: / / arxiv.org / abs / 2203.06061 (2022).

[0160] 17. Pierangeli, D. et al. Living optical random neural network with three dimensional tumor spheroids for cancer morphodynamics. Commun Phys 3, 1-10 (2020). 18. Spall, J., Guo, X., Barrett, T. D. & Lvovsky, A. I. Fully reconfigurable coherent optical vector-matrix multiplication. Opt. Lett. 45, 5752 (2020).

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Claims

1. CLAIMS1. An optical multiplication system comprising: an optical input arrangement configured to provide a plurality of input beams of light encoding an input vector; a beam replication arrangement configured to receive the plurality of input beams and produce a plurality of spatially-separated replica beams of each input beam; a modulator comprising a plurality of modulator elements configured to encode values of a matrix, wherein each modulator element is configured to modulate one of the replica beams received from the beam replication arrangement according to the respective encoded matrix value; and a light-summing optical arrangement configured to converge the modulated replica beams output from the modulator elements to encode an output vector representing a result of multiplying the matrix by the input vector, wherein the optical input arrangement comprises a three-dimensional light-guide providing the plurality of input beams to the beam replication arrangement as a two- dimensional array of input beams.

2. The system of claim 1, wherein the three-dimensional light guide comprises a two- dimensional array of elongate light-guiding elements.

3. The system of claim 2, wherein the light-guiding elements comprise optical fibres.

4. The system of claim 2 or 3, wherein the light guiding elements comprise photonic waveguides.

5. The system of claim 3, wherein the light guiding elements further comprise photonic waveguides connected to the optical fibres, the photonic waveguides configured to reduce a spacing of the light guiding elements in a plane perpendicular to a length of the light guiding elements.

6. The system of any of claims 2 to 5, wherein the array of elongate light-guiding elements25is a square array or a hexagonal array.

7. The system of any preceding claim, wherein each input beam is provided as a single transverse mode, optionally a TEMoo Gaussian mode.

8. The system of any of claims 1 to 6, wherein each input beam comprises a plurality of transverse modes.

9. The system of any preceding claim, wherein the optical input arrangement comprises: i) a plurality of modulated optical emitters configured to produce the plurality of input beams; or ii) one or more light sources and a plurality of optical modulators configured to modulate light from the one or more light sources to produce the plurality of input beams.

10. The system of any preceding claim, wherein the optical input arrangement is configured to encode the input vector using temporal modulation of the input beams.

11. The system of any preceding claim, wherein the optical input arrangement is configured to encode each element of the input vector using plural input beams.

12. The system of claim 11, wherein the plural input beams corresponding to the same element of the input vector have different intensities.

13. The system of claim 12, wherein the intensities of the plural input beams corresponding to the same element of the input vector are related to one another by an exponential decay function.

14. The system of any preceding claim, wherein the beam replication arrangement comprises a diffractive optical element.

15. The system of any of claims 1 to 13, wherein the beam replication arrangement comprises a pair of micro-lens arrays arranged in series with a focusing lens.

16. The system of any preceding claim, wherein the beam replication arrangement is configured to produce the plurality of spatially-separated replica beams of each input beam as a two-dimensional array of replica beams.

17. The system of any preceding claim, wherein the modulator comprises a spatial light modulator, for example a liquid-crystal spatial light modulator.

18. The system of any preceding claim, wherein the modulator comprises a digital micromirror device.

19. The system of any preceding claim, wherein the modulator comprises: a first component configured to change a modulation applied to the replica beams on a first timescale; and a second component configured to change a modulation applied to the replica beams on a second timescale that is longer than the first timescale, optionally wherein the modulation applied to the replica beams by the second component is static.

20. The system of any preceding claim, wherein the light-summing arrangement comprises a microlens array, and optionally further comprises a spherical lens.

21. The system of any preceding claim, wherein the light-summing arrangement is configured to encode the output vector as a plurality of output beams encoding elements of the output vector.

22. The system of claim 21, wherein the optical multiplication system further comprises an optical output arrangement configured to receive the plurality of output beams and produce an output corresponding to the output vector based on the plurality of output beams.

23. The system of claim 22, wherein the optical output arrangement comprises a three- dimensional light-guide configured to receive the plurality of output beams.

24. The system of claim 23, wherein the three-dimensional light guide comprises a two- dimensional array of elongate light-guiding elements, optionally wherein the light-guiding elements comprise optical fibres, optionally large-mode area optical fibres.

25. The system of any of claims 22 to 24, wherein the optical output arrangement comprises a detector arrangement configured to individually read out each of the plurality of output beams.

26. A method of performing analogue computing, comprising: encoding an input vector using a plurality of input beams of light; producing a plurality of spatially-separated replica beams of each input beam; modulating the replica beams using a plurality of modulator elements encoding a matrix according to the respective encoded matrix value of each modulator element; and converging the modulated replica beams to encode an output vector representing a result of multiplying the matrix by the input vector, wherein the plurality of input beams is provided as a two-dimensional array of input beams.

27. The method of claim 26, wherein the replica beams are produced using a diffractive optical element.

28. The method of claim 26 or 27, further comprising producing an output corresponding to the output vector based on the converged beams.28

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