Optical computing system for solving partial differential equations

Abstract

An optical computing system for solving partial differential equations, comprising: a degenerate cavity laser (DCL); an imaging system within an optical cavity of the DCL, the DCL configured to generate a laser beam which performs a number of round trips within the optical cavity; one or more optical elements encoding mathematical operators of the PDE and configured to change at least one physical characteristic of the beam; a memory and at least one computer processor configured to: receive a PDE; perform a first association associating a spatial coordinate grid with a Fourier plane of the imaging system; perform a second association associating a time coordinate with a number of round trips of the beam; output a PDE solution by measuring the electric field of the beam, wherein a complex amplitude and phase of the measured electric field is converted to the solution based on the first and second associations.

Description

P-641241 -PCOPTICAL COMPUTING SYSTEM FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONSFIELD OF THE INVENTION

[0001] The present invention relates generally to the field of electro-optics, and more particularly to an optical computing system configured to solve partial differential equations by optical means.BACKGROUND OF THE INVENTION

[0002] Partial differential equations (PDEs) describe and model the behaviour of physical systems. PDEs are multivariable and generally difficult to solve. As such, there may be a desire in the art for a technique and a system to solve PDEs optically, in particular by means of an optical computing system which is programmable and configurable for a variety of operators.

[0003] Degenerate cavity lasers (DCLs) are a class of optical resonators designed to support a large number of transverse laser modes that resonate with identical frequencies. This characteristic enables manipulation of the spatial properties of laser beams, including the control of spatial structures and spatial frequencies of light. In DCLs, optical elements can be placed both in the image plane or Fourier / conjugate planes between lenses, allowing for control over the physical properties of a laser beam, and more specifically, the coupling between the different laser modes or different laser beams.

[0004] Digital degenerate cavity laser systems can offer precise control over beam characteristics such as intensity, phase, polarization and coherence, as well as the interaction between the different modes, thereby allowing for the creation of intricate and versatile light patterns.

[0005] Phase masks, including both static and programmable types like Diffractive Optical Elements (DOEs) and Spatial Light Modulators (SLMs), can contribute to controlling the physical properties of light, and of interactions between the different modes thereof, in advanced optical systems. Static phase masks, such as DOEs, can be used to imprint a fixed phase pattern onto the incident light, enabling applications like beam shaping, splitting, or focusing. Programmable phase masks, such as SLMs, offer configurable control over the phase distribution of the beam, allowing real-time manipulation and reconfiguration of the light’s spatial properties.P-641241 -PCBRIEF SUMMARY OF THE INVENTION

[0006] According to one or more embodiments of the invention, there is provided n optical computing system for solving partial differential equations (PDEs), comprising: a degenerate cavity laser (DCL) comprising an imaging system located within an optical cavity of the DCL, the DCL configured to generate a first laser beam which iteratively performs a number of round trip propagations within the optical cavity; one or more optical elements, located at one or more optical planes of the imaging system, configured to change at least one physical characteristic of the first laser beam, the one or more optical elements selectively configured to encode one or more mathematical operators of the PDE; at least one computer processor; and a memory containing instructions which, when executed by the at least one computer processor, cause the at least one computer processor to: receive a PDE to be solved; perform a first association associating a spatial coordinate grid with an image plane of the imaging system; perform a second association associating a time coordinate with a number of round trip propagations of the first laser beam; and output a solution of the PDE at a desired time by measuring the electric field of the first laser beam after a number of round trip propagations corresponding to the desired time, wherein a complex amplitude and phase of the measured electric field is converted to the solution based on the first and second associations.

[0007] According to some embodiments, the system may further comprise a unidirectional coupled laser, wherein boundary conditions are enforced by the unidirectional coupling.

[0008] According to some embodiments, boundary conditions are enforced by controlling one of the one or more optical elements to inject a second laser beam with known physical characteristics at a position in the image plane associated with a boundary condition.

[0009] According to some embodiments, one or more optical elements are chosen from the list including: diffractive optical elements; spatial light modulators; beam splitters; beam combiners; lenses; mirrors; gain media; wave plates; digital micromirror devices; and / or nonlinear optical materials; and the physical characteristics of a laser beam include at least one of: phase; amplitude; polarization; spatial profile; direction of propagation; spectral content; and number of split beams.

[0010] According to some embodiments, the imaging system comprises one or multiple 4f telescopes, wherein the multiple 4f telescopes are connected in series or in parallel.

[0011] According to some embodiments, an optical element of the one or more optical elements is configured to split the first laser beam into a number of laser beams corresponding to a number of mathematical terms of the PDE.P-641241 -PC

[0012] According to some embodiments, an optical element located at an optical plane of the imaging system is configured to change the phase or amplitude of a laser beam, the change corresponding to a scalar factor in the PDE.

[0013] According to some embodiments, at least some of the one or more optical elements are located at consecutive optical planes in the optical cavity, corresponding to a composition of functions in the PDE.

[0014] According to some embodiments, the one or more optical elements comprises a nonlinear optical material corresponding to a nonlinear operation in the PDE.

[0015] According to some embodiments, the system further comprises a number of optical cavities corresponding to the number of dependent variables of the PDE, wherein the optical cavities are connected according to the PDE.

[0016] According to some embodiments, coupled lasers are configured to represent a spatial variable of the PDE.

[0017] According to some embodiments, a split laser beam is configured to represent a spatial operator of the PDE.

[0018] According to some embodiments, the system further comprises a camera configured to capture interference patterns between spots of the same laser beam.

[0019] According to some embodiments, the desired time is the time taken for the system to reach a steady state and the solution is the steady -state solution.

[0020] According to some embodiments, there is provided a method for optically computing solutions of partial differential equations (PDEs), the method comprising: receiving a PDE to be solved; generating a first laser beam which iteratively performs a number of round trip propagations within an optical cavity of a degenerate cavity laser (DCL); changing at least one physical characteristic of the first laser beam using one or more optical elements located at one or more optical planes of an imaging system located within the optical cavity; performing a first association associating a spatial coordinate grid with an image plane of the imaging system; performing a second association associating a time coordinate with a number of round trip propagations of the first laser beam; and outputting a solution of the PDE at a desired time by measuring the electric field of the first laser beam after a number of round trip propagations corresponding to the desired time, wherein a complex amplitude and phase of the measured electric field is converted to the solution based on the first and second association, wherein the one or more optical elements are selectively configured to encode one or more mathematical operators of the PDE.P-641241 -PC

[0021] According to some embodiments, the method further comprises enforcing boundary conditions using unidirectional coupling.

[0022] According to some embodiments, the method further comprises enforcing boundary conditions using one of the one or more optical elements to inject a second laser beam with known physical characteristics at a position in the image plane associated with a boundary condition.

[0023] According to some embodiments, the one or more optical elements are chosen from the list comprising: diffractive optical elements; spatial light modulators; beam splitters; beam combiners; lenses; mirrors; gain media; wave plates; digital micromirror devices; and / or nonlinear optical materials; and the physical characteristics of a laser beam include at least one of: phase; amplitude; polarization; spatial profile; direction of propagation; spectral content; and number of split beams.

[0024] According to some embodiments, the method further comprises adding to the imaging system one or multiple 4f telescopes, wherein the multiple 4f telescope are connected in series or in parallel.

[0025] According to some embodiments, the method further comprises using an optical element of the one or more optical elements to split the first laser beam into a number of laser beams corresponding to a number of mathematical terms of the PDE.

[0026] According to some embodiments, the method further comprises using an optical element located at an optical plane of the imaging system to change the phase or amplitude of a laser beam, the change corresponding to a scalar factor in the PDE.

[0027] According to some embodiments, the method further comprises configuring at least some of the one or more optical elements at consecutive optical planes in the optical cavity, corresponding to a composition of functions in the PDE.

[0028] According to some embodiments, the method further comprises using a nonlinear optical material, corresponding to a nonlinear operation in the PDE.

[0029] According to some embodiments, the method further comprises adding a number of optical cavities corresponding to the number of dependent variables of the PDE; and connecting the optical cavities according to the PDE.

[0030] According to some embodiments, the method further comprises configuring coupled lasers to represent a spatial variable of the PDE.

[0031] According to some embodiments, the method further comprises configuring a split laser beam to represent a spatial operator of the PDE.P-641241 -PC

[0032] According to some embodiments, the method further comprises capturing interference patterns between neighbouring laser beams.

[0033] According to some embodiments, the desired time is the time taken for the system to reach a steady state and the solution is the steady -state solution.BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Non-limiting examples of embodiments of the disclosure are described below with reference to figures attached hereto. Dimensions of features shown in the figures are chosen for convenience and clarity of presentation and are not necessarily shown to scale. The subject matter regarded as the invention is particularly pointed out and distinctly claimed in the concluding portion of the specification. The invention, however, both as to organization and method of operation, together with objects, features, and advantages thereof, may be understood by reference to the following detailed description when read with the accompanied drawings. Embodiments are illustrated without limitation in the figures, in which like reference numerals indicate corresponding, analogous, or similar elements, and in which:

[0035] Fig. 1 is a schematic drawing of a degenerate cavity laser, according to some embodiments of the invention;

[0036] Fig. 2 is a schematic drawing of an example module for injecting boundary conditions into an optical cavity, according to some embodiments of the invention;

[0037] Fig. 3 is a schematic drawing of a degenerate cavity laser, according to some embodiments of the invention;

[0038] Fig. 4 is a flowchart of a method for optically computing solutions of partial differential equations (PDEs), according to some embodiments of the invention; and

[0039] Fig. 5 is a block diagram of an exemplary computing device which may be used with some embodiments of the invention.

[0040] It will be appreciated that for simplicity and clarity of illustration, elements shown in the figures have not necessarily been drawn to scale. For example, the dimensions of some of the elements may be exaggerated relative to other elements for clarity. Further, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements.P-641241 -PCDETAILED DESCRIPTION OF THE INVENTION

[0041] In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the invention. However, it will be understood by those skilled in the art that the present invention can be practiced without these specific details. In other instances, well-known methods, procedures, components, modules, units and / or circuits have not been described in detail so as not to obscure the invention.

[0042] Some features or elements described with respect to one embodiment may be combined with features or elements described with respect to other embodiments. For the sake of clarity, discussion of same or similar features or elements may not be repeated.

[0043] According to some embodiments of the invention, there is provided an optical computing system capable of solving a wide range of partial differential equations (PDEs) by utilizing the physical dynamics of a degenerate cavity laser (DCL) system integrated with static and programmable phase and / or amplitude masks and other optical elements. By associating variables of the PDE with the physical properties of the laser and mapping mathematical operations and / or operators to optical processes within the cavity, some embodiments of the invention provide an approach to fast computation of complex mathematical equations, which may far surpass the speed and accuracy of traditional electronic computing methods.

[0044] Embodiments of the invention are distinct from a mere mathematical method in that mathematical operators and terms are embedded, encoded, converted, translated, or otherwise associated with physical elements (such as optical elements) which affect, in a technical manner, the physical characteristics of a laser beam, from which a solution can be derived by measuring the laser beam, as described further herein.

[0045] Some embodiments of the invention may be provided as a device, apparatus, or system, which may be referred to as an optical computer, programmable optical coupler, or optical computing system, that finds the evolution in time and / or finds the steady state solutions of mathematical problems that can be represented as a partial differential equation (PDEs) and / or system (e.g. set) of PDEs. The operating principle of the optical computing system is to embed (e.g. encode, convert, translate, associate) the mathematical problem into a laser cavity and utilize the evolution of the laser state to “compute” either the dynamical evolution, e.g., the time-dependent behaviour of mathematical variables in the equation(s), or the steady state solution.

[0046] Fig. 1 shows a schematic diagram of a programmable optical computing device configured to solve the two-dimensional heat diffusion PDE problem. The optical computing device may be in communication with a computer processor (e.g. a non-optical based computer processor) whichP-641241 -PCreceives a PDE problem. For example the computer processor may receive one or more PDEs to be solved. The computer processor may also receive one or more initial and / or boundary conditions.d U (x y t)

[0047] The optical computing device may be configured to find the steady state, — = 0, of thefollowing equation (the heat diffusion equation in two dimensions):dU x,y, t)∂U(x,y,t) / ∂t = κ∇2U (x, y, t) + ρ (x, y) (1) with boundary conditions U((x, y) ∈ domain boundary, t) = Uedge(2) and initial conditions U(x,y,t = 0) = U0(x,y), (3) where (x,y) is a distribution of sources and sinks in space, and K is a uniform heat diffusion coefficient. Note that this can be generalized to a non-uniform heat diffusion coefficient in a straightforward way.

[0048] The mathematical components of the PDE problem may be embedded, encoded, converted, associated, translated, or otherwise mapped into the optical and physical properties of an imaging system of the programmable optical computing device. The mathematical components to be implemented in the imaging system may comprise: the spatial coordinate grid (x,y) onto which the problem is defined; the time coordinate t; the value of the unknown variable U at every point in the spatial grid and time coordinate; the time derivative operation ∂ / ∂t; the Laplace operator ∇², and the heat sources and / or sinks ρ(x,y).

[0049] The Laplace operator is the sum of second order spatial derivatives. In this example, theLaplace operator is of the form ∇2= ∂2U / ∂x2+ ∂2U / ∂y2. The Laplace operator can be of any dimension. Forexample, for PDEs that model physical phenomena it may be useful to use a three-dimensional Laplacian.

[0050] The optical computing system may comprise a degenerate cavity laser (DCL) 100. The DCL comprises an optical cavity 100a. The DCL may also comprise a means of generating a laser beam. For example, the DCL may comprise a gain medium 101 to generate a laser beam 150.

[0051] Although it will be understood that there may be a plurality of modes of a laser beam within the optical cavity the modes will be referred to collectively herein as being of the same laser beam (e.g. first laser beam). Use of “second” laser beam(s) may refer to laser beam(s) with different physical characteristics for a different purpose (e.g. imposing boundary conditions, in some embodiments), which may be generated by or originate from a different source.

[0052] The initial and / or boundary conditions of the PDE problem may be encoded into the physical characteristics of a laser beam (e.g. a first and / or second laser beam). The physical characteristics mayP-641241 -PCinclude, for example: phase; amplitude; polarization; spatial profile; direction of propagation; spectral content; and number of split beams. A split beam will be understood as a beam split from an originating beam; a split beam may be referred to as a mode, channel.

[0053] The initial conditions of the PDE problem may be implemented by propagating the beam 150, generated by gain medium 101, through a first optical cavity comprising optical elements configured such that the beam 150 will reach a steady state corresponding to the initial conditions. Once this steady state has been reached, the beam 150 may then be propagated within a second optical cavity with an imaging system comprising optical elements selected and / or configured according to the mathematical terms of the PDE.

[0054] The laser beam may then travel through an imaging system comprised in the DCL (e.g. located in the optical cavity 100a). The optical cavity 100a may further comprise mirrors 103 to allow the laser beam to iteratively circulate or propagate through the cavity (e.g. repeat a number of round trips or cycles). This is shown schematically by the arrows on the laser beam forming a loop, and by the circular arrows in the centre of the optical cavity 100a indicating a number N of loops, propagations, round trips, or cycles.

[0055] An imaging system may include an arrangement of optical elements forming a series of optical planes including: image planes; Fourier planes, conjugate, planes; and / or other intermediate planes. When a laser beam (e.g. beam 150) travels through a Fourier plane, it undergoes an effective Fourier transform. Various optical elements may be placed, located, and / or configured at any of the image, Fourier, conjugate and / or intermediate planes to effect certain transformations on the laser beam, e.g. changing its physical characteristics, corresponding to variables and operators in the PDE problem.

[0056] In some embodiments, the imaging system may comprise a 4f telescope 120, which comprises two lenses, 110a and 110b, with a spacing of 2f between them, where f is the focal length of the lenses. Between the two lenses, at their common focal plane, lies a Fourier plane 122.

[0057] Example optical elements which may be used with some embodiments of the invention include: diffractive optical elements (DOE); spatial light modulators (SLM); beam splitters (e.g. polarizing beam splitters); beam combiners; lenses (e.g. 4f telescope arrangement); mirrors; gain media; wave plates (e.g. quarter-wave plates); digital micromirror devices (DMD); and / or nonlinear optical materials.

[0058] A non-linear optical material may be any material which has non-linear optical -physical behaviour, such as saturable absorption, frequency doubling, or Kerr non-linearity. For example, this could be graphene, which has saturable absorption properties, or a crystal whose internal structure doubles the frequency of incoming light, such as KTP (Potassium titanyl phosphate).P-641241 -PC

[0059] Fig. 1 shows an embodiment of the optical computing device where the imaging system comprises a series of cascaded or successive 4f telescopes, (in this embodiment, connected in series). Each 4f telescope maps an image plane to an image plane of a neighbouring telescope, with each 4f telescope comprising a Fourier plane between its two image planes. Alternatively, depending on the frame of reference, the 4f telescopes may map neighbouring Fourier planes with an image plane in the middle.

[0060] In Fig. 1, a first image plane 121 may comprise the gain medium 101 which emits the laser beam 150. The angled plane comprising mirror 103 may be approximated as a first Fourier plane. A second image plane 122 may comprise an optical element 200, which may, for example, enforce the boundary conditions on the laser, as shown in Fig. 2. A second Fourier plane 124 may comprise an optical element such as a phase mask 104 corresponding to the Laplace operator, such that when the laser beam 150 passes through the plane the physical characteristics of the laser change, according to an effective implementation of the Laplace operator. Another mirror 103a lies on an approximate image plane.

[0061] In some embodiments, for example as discussed above, the 4f telescopes are connected in series. According to other embodiments, the 4f telescopes may be connected in parallel. According to some embodiments, the system may comprise a single 4f telescope. It will thus be understood that an optical computing system according to embodiments of the invention can include one or more 4f telescopes, connected in any desired combination.

[0062] The phase mask 104 may be any optical element configured to change the phase of incoming light. According to some embodiments, the phase mask may be a spatial light modulator (SLM). According to some embodiments, the phase mask may be a diffractive optical element (DOE). A phase mask may be programmable, for example being capable of being digitally adjusted to change the phase of light passing through it.

[0063] The image planes, such as planes 121 and 122, correspond to the spatial coordinate grid on which the problem is defined. In particular, a point (x, y) of the PDE problem can be associated (e.g. first association) with a spatial position on an image plane of a 4f telescope. A computer processor (e.g. a non-optical based computer) may be configured to map the spatial coordinate grid of the problem onto the image planes. This may comprise configuring the computer processor to compute coordinate transforms between the spatial coordinate grid and the planes.

[0064] According to some embodiments, the spatial variables, for example x and y, may be represented by coupled lasers. For example, the Laplace operator can be represented by couplingP-641241 -PCneighbouring lasers (e.g. implemented by an appropriate optical element) throughout the evolution of the system.

[0065] Fourier planes, for example plane 124, correspond to a spatial frequency grid with points (kx, ky) where k = 2π / wavelength is the wavenumber.v y' wavelength

[0066] The time coordinate t of the PDE problem can be associated (e.g. second association) by the processor to a number of iterative round trip propagations carried out by the laser beam in the cavity. For example, at t = 0 the laser is emitted, or otherwise injected into the cavity, and time elapsed thereafter is measured in units of “cavity -length divided by the speed of light”. For example, t = 3 corresponds to the completion of the third circulation by the laser beam, with the time elapsed given cavity lengthby 3 • seconds.speed of light

[0067] A computer processor may use data obtained from imaging techniques, for example using streak cameras and / or photo-diodes, to capture the time evolution of the laser, such as detecting when circulations / propagations around the cavity are completed. The cavity length may be manually defined by the user or automatically computed by the computer processor according to the chosen implementation, e.g. depending on the number of optical elements and / or cavities used.

[0068] The laser beam will evolve in time. According to some embodiments, depending on the PDE problem, the laser beam may reach a steady state (e.g. time derivative equal to zero). For example, the diffusion equation (1) will reach a steady state. The laser beam has reached a steady state when its optical properties no longer change with time. This steady state can then be read and converted to the steady-state solution for the diffusion equation (1). Conversion to a readable solution, e.g. human or machine readable solution (as opposed to measured properties of the laser beam encoding a solution), may be made by converting a complex (e.g. imaginary in the mathematical sense of i or √−1 ) field, containing amplitude and / or phase, based on the first and second associations. For example, in this embodiment, the complex field may be the measured electric field.

[0069] The laser beam 150 propagating through the optical cavity 100a (e.g. N times as shown by circular arrows) has an associated electric field. The electric field takes on various values as the laser beam passes through the various optical planes. In particular, in some embodiments, the value of U(x,y, t) is given by the complex phase and / or amplitude of the electric field at the corresponding point (x, y) in the image plane at a time t.

[0070] These properties can be read / measured, e.g. using a suitable detector or measurement module, after any number of trips N and converted to provide a solution at associated / corresponding time t. Alternatively, these properties may be read after a steady state has been reached.P-641241 -PC

[0071] An isolator 170, such as an optical diode, may prevent b ackpropagation of laser beam 150, which may introduce unwanted feedback which affects the beam evolution and consequently the PDE solution.

[0072] Fig. 2 shows an example of how the boundary conditions (2) may be implemented. The optical element 200 may comprise a polarizing beam splitter 202, a quarter-wave plate 203 and a digital micromirror device (DMD) 204. The laser beam 150 passes through the input plane 201 and a second laser beam 250 is injected into the cavity. Each pixel in the DMD 204 has an “on” state where it reflects light from the input plane 201, e.g. the laser beam 150, to the output plane 205, and an “ofi” state where it reflects light from a side source, e.g. the second laser beam 250, to the output plane. The laser beam 150 is rotated from its linearly-polarized state by optical element 200 in order to achieve high transmission.

[0073] By placing this optical element 200 at an image plane, the second laser beam 250, with known physical characteristics, can be injected at certain positions, corresponding to the spatial grid, in the plane according to the given boundary conditions (3) of the PDE problem. Optical element 200 splits the second laser beam 250 into a number of non-changing beams which replace laser beam 150 at the boundaries, thereby fixing the value of the electric field at the boundaries and thus enforcing the boundary conditions. For example, Dirichlet boundary conditions may be implemented in this way.

[0074] In other embodiments, Dirichlet boundary conditions may be implemented by coupling boundary spots to one another to realize the specific boundary conditions, and coupling boundary spots to non-boundary spots (bulk spots) unidirectionally, whereby boundary spots are coupled to bulk spots but not vice versa. Neumann boundary conditions may be implemented by controlling the self-coupling of boundary lasers, as well as coupling with adjacent bulk spots, according to the value of the given boundary condition. As used herein, “spots” may be used to refer to the spatial modes of the same laser beam.

[0075] According to some embodiments, the second laser beam 250 may be generated by an external source outside of the DCL. Alternatively, the second laser beam 250 may be generated within the DCL. According to some embodiments, the second laser beam is generated by a gain medium.

[0076] In the laser cavity of Fig. 1, the governing equations for the electric field E may be written as E = (>.5(1 — s) aeG— + <zeG0.5-\ / s[ / < V2+ ip(x,y)]E, (4) 1 G = — P — G 1 + (5)iiP-641241 -PCwhere G is gain, a are losses, s is the beam-splitting factor, P is the active medium pump, Isatis the saturation intensity of the gain medium, and TCis the fluorescence lifetime of the gain medium. E and G represent, respectively, the time derivatives of the electric field and gain.

[0077] Alternatively, the governing equations may be written in the following discrete form:E1= 70.5(1 - s)+ O.^aecT* + E;+1.„ + E^ + E"K+1- lEfr) + iPl.kE"„], <>n+l pnb— Cj,knTCbj,k

[0078] Note that the sources / sinks p(x, y) are implemented by introducing complex self-coupling within a single laser beam. According to some embodiments, a spatial operator may be represented by a split laser beam. For example, the Laplace operator may be implemented by splitting a beam into two beams, wherein the first split beam undergoes coupling with other nearest neighbour laser beams, and the second split beam undergoes complex self-coupling and recombining.

[0079] This formulation may be extended to non-uniform heat diffusion coefficient K(X, y). In this case, extra coupling terms may be introduced accordingly.

[0080] If the electric field is rewritten as E = eA+i^, where A and <p are real functions of (x, y, t) and represent, respectively, the amplitude and phase of the electric field, the following equation for the phase, in particular its time derivative <p, is obtained:0(x,y, t) = <zeG(0.5-\ / s[2Vi4 • V0 + KV20 + p(x,y)]). (6)

[0081] For uniform amplitudes (V / 1 = 0, e.g. a condition enforced by the gain dynamics), the equation for the phase (5), wherein the value of the phase is obtained by measuring E (x, y, t) as described above, gives the solution U (x, y, t) to the heat diffusion equation (1), with aeG0. Ss s acting as a time scale for the equation. Thus, the optical computing system, according to some embodiments of the invention, can solve the two-dimensional heat diffusion PDE problem (1) - (3).

[0082] The discrete form of (6), for uniform amplitudes, small angles and in the steady state, is given by:(. steady state.. steady state.. steady state.. steady state.. steady states.nMoA+ 07+i,fc + 0U-1 + 0M+1 ) + Pi,k = 0 (6a)

[0083] The governing equations, e.g. (4) - (6), may be supplied (e.g. input) to a computer processor to enable it to compute the solution. They may be stored in memory in some embodiments.

[0084] In addition to finding the steady state solution, it is also possible to find the time-dependent evolution of PDEs. The evolution of the laser field in the DCL is identical to the evolution of the variables in the PDE. Imaging techniques can therefore be used to capture the time dynamics of theP-641241 -PClaser which can then be converted into the time evolution of the mathematical components of the PDE.

[0085] For example, a camera may be used to capture interference patterns of neighbouring laser beams, and / or between spots of the same laser beam These interference patterns may be read by a computer processor and converted into time-dependent solutions.

[0086] Fig. 3 shows a schematic diagram of a programmable optical computing device 300 comprising a DCL 300a with an imaging system comprising a series of cascaded or successive 4f telescopes 320. The 4f telescope 320 comprises lenses (e.g. 310a) and optical planes (e.g. 322 and 324).

[0087] A laser beam 350 is produced by a gain medium 301 and propagates around the cavity, for example directed by mirrors 303. The laser beam passes through a boundary condition module 307. In some embodiments, this may be optical element 200 shown in Fig. 2; according to other embodiments, the boundary condition module 307 may implement unidirectional coupling as described herein.

[0088] The optical computing device may also comprise a beam sampler 360 and a measurement module 380 in order to collect measurements, for example measuring the physical characteristics of the laser beam after N propagations. Measurement module 380 may comprise streak cameras and / or photo-diodes. Measurement module 380 may comprise a camera configured to capture interference patterns between spots of the same laser beam.

[0089] The DCL may comprise a beam splitter 302a and a beam combiner 302b. Beam splitter 302a may selectively (e.g. in a programmable manner, as controlled by a computer processor) split laser beam 350 into two laser beams 350a and 350b which follow different optical paths. Split beams 350a and 350b may be recombined by beam combiner 302b.

[0090] An isolator 370, such as an optical diode, may prevent backpropagation of laser beam(s) within the cavity, which may introduce unwanted feedback which affects the beam evolution and consequently the PDE solution.

[0091] According to some embodiments, beam splitters and beam combiners as arranged in optical computing device 300 may be utilised to implement mathematical addition present in the PDE. A beam splitter may be configured to split a laser beam (e.g. 350) into a number of laser beams corresponding to a number of mathematical terms of the PDE. The emergent laser beams (e.g. 350a and 350b) represent the terms to be summed. For example, in order to compute V2(7 + U, a laser beam may be split into two laser beams, one for each term (one beam corresponding to the U in V2[ / P-641241 -PCand the other corresponding to the term +( / ). The first split beam may then propagate through further optical elements to implement the operation V2[7.

[0092] As the split beams undergo different transformations, their optical path lengths diverge, resulting in the beams having different phases. This is mathematically equivalent to multiplying the corresponding terms in the PDE by a complex number with an amplitude of 1. The relative lengths of the optical paths may be adjusted or controlled according to the phase of the complex number in order to implement addition and subtraction.

[0093] For example, adding two terms of the PDE corresponds to their respective laser beams having a phase difference of 2n / r, where n is an integer, resulting in constructive interference. This is mathematically equivalent to multiplying the added terms of the PDE by e^2nn^1= 1.

[0094] Subtracting a term is equivalent to multiplying it by a scalar factor of e2(n+z)’ri= — 1. This translates to its corresponding laser beam having a phase difference of 2 (n + n relative to the laser beams corresponding to positive terms of the PDE, as this phase difference results in destructive interference.

[0095] The result of the sum may be obtained by recombining the beams with a beam combiner (e.g.302b).

[0096] Those knowledgeable in the art will be able to understand from this disclosure how optical computing devices 100 and 300 can be modified and / or generalized to implement various PDEs, for example other than the diffusion equation. For other PDEs, embodiments of the optical computing device implementing additional operators and other mathematical elements of the PDE can be executed as follows.

[0097] According to some embodiments, scalar multiplication may be achieved by placing an optical element configured to change the phase and / or amplitude of a laser beam at an image plane of an imaging system. As the laser beam passes through the optical element, its phase and / or amplitude will change, effectively corresponding to multiplication by a scalar factor. For example, an SLM can be configured to change the amplitude, or intensity, of the laser beam by a scalar factor, for example a factor of 0.5 corresponding to a multiplicative operation such as - U.

[0098] In order to increase the amplitude of the laser beam, e.g. for an operation such as 2U, an optical element such as a lens may be used. For example, the lens may focus the laser beam into half of its diameter, corresponding to a two-fold increase in the amplitude and thus achieving 2U. Alternatively, a gain medium may also be used to implement 2U. In some embodiments, 2U may be implemented with a SLM by rescaling the PDE problem.P-641241 -PC

[0099] According to some embodiments, in particular those where the given PDE problem contains multiple coupled unknown dependent variables, the optical computing device may comprise a number of optical cavities corresponding to the number of dependent variables of the PDE. A beam splitter may split a laser beam into a number of laser beams corresponding to the number of terms in the problem; each split beam may subsequently be directed, for example using mirrors, towards a cavity corresponding to the relevant variable. The multiple cavities may be connected according to the terms and operators of the PDE problem. For example, non-linear multivariable terms, such as U2V, may be implemented by connecting the corresponding cavities with a non-linear material. The split laser beams may then be added and recombined using the addition method described herein. As used herein, connecting according to the PDE will thus be understood as having a physical arrangement (e.g. series, parallel) of cavities selected to implement a desired corresponding mathematical effect determined by the given PDE to be solved.

[0100] According to some embodiments, optical elements may be placed at consecutive planes within the DCL, each optical element corresponding to a particular mathematical function or operation. As a laser beam travels through each plane, it can undergo cumulative changes to its physical properties / characteristics, as influenced by the optical elements. This corresponds to composition of functions. According to some embodiments, this may be achieved by configuring the laser to operate in a unidirectional ring arrangement.

[0101] A conjugate plane refers to a plane whose points have a one-to-one mapping with another plane. According to some embodiments, an operator may be encoded into a conjugate Fourier plane, corresponding to a non-local, spatially translational-invariant operator, for example a Fourier transform or Laplace transform. In some embodiments, the operator may be encoded into an image plane, in which case the operator is location-specific, such as a differential operator (e.g. Laplacian).

[0102] According to some embodiments, splitting and combining of lasers may occur in consecutive image planes. When a laser is split and recombined, some of the light may scatter away from the desired path. Some of the split beams may therefore be blocked to prevent scattered light from propagating within the cavity and affecting the solution of the PDE.

[0103] An isolator, such as an optical diode, may prevent backpropagation of laser beam(s) within the cavity, which may introduce unwanted feedback which affects the beam evolution and consequently the PDE solution.

[0104] According to some embodiments, a nonlinear optical material may be placed in the appropriate Fourier plane within the cavity in order to carry out nonlinear operations. For example, introducing a frequency doubling crystal in an image plane may correspond to the operation U — U2.P-641241 -PC

[0105] Further embodiments implementing other mathematical operations can be achieved by analysing the components of the PDE problem and correspondingly configuring and / or introducing additional optical elements which produce a result on the physical characteristics of the laser beam analogous to the mathematical operation. The one or more optical elements used by embodiments of the invention may be selectively configured to encode one or more mathematical operators of the PDE. For example, optical elements may be programmable by a computer processor to change their properties, e.g. optical thickness, mask pattern, or the like. Optical elements may be selectively inserted, for example manually, or by an actuation means controlled by a computer processor.

[0106] Fig. 4 shows a flowchart of a method 400 for optically computing solutions of partial differential equations, according to some embodiments of the invention.

[0107] Method 400 includes receiving (Step 410) a PDE to be solved. According to some embodiments, Step 410 may also include receiving boundary conditions.

[0108] Method 400 includes generating (Step 420) a first laser beam to iteratively perform a number of round-trip propagations within an optical cavity of a degenerate cavity laser (DCL). For example, an optical computing system having a degenerate cavity laser 100 as described above may be used.

[0109] Method 400 further includes changing (Step 430) at least one physical characteristic of the first laser beam using one or more optical elements located at one or more optical planes of an imaging system located within the optical cavity. The optical elements may be as described as herein, for example, chosen from the list comprising: diffractive optical elements; spatial light modulators; beam splitters; beam combiners; lenses (e.g. in particular a 4f telescope arrangement); mirrors; gain media; wave plates; digital micromirror devices; and / or nonlinear optical materials. The physical characteristics may be as described herein, and may include at least one of: phase; amplitude; polarization; spatial profile; direction of propagation; spectral content; and / or number of split beams.

[0110] The one or more optical elements are selectively configured to encode one or more mathematical operators of the PDE. Optical elements may be selected or configured for particular mathematical operators as described herein. A connected computer processor may selectively configure one or more optical components for the intended mathematical operator. In some embodiments, optical elements may be added manually and / or automatically. For example, an optical element, such as a lens, may be actuated (e.g. swivelled, inserted, lowered, moved) by an actuating means (e.g. servomotor) into an appropriate position to provide an appropriate optical element for changing the physical characteristics of the laser beam. In some embodiments, programmable optical elements may be pre-included in the optical cavity but programmed so as not to affect the laser beam if not required (e.g. they may be turned off and allow the laser to pass through unimpeded).P-641241 -PC

[0111] Method 400 may include performing a first association (Step 440) associating a spatial coordinate grid with an image plane of the imaging system. The association may be performed by a connected computer processor.

[0112] Method 400 may include performing a second association (Step 450) associating a time coordinate with a number of round trip propagations of the first laser beam. The association may be performed by a connected computer processor.

[0113] Method 400 may include outputting (Step 460) a solution of the PDE at a desired time by measuring the electric field of the first laser beam after a number of round trip propagations corresponding to the desired time, wherein a complex amplitude and / or phase of the measured electric field is converted to the solution based on the first and second association, as described herein. According to some embodiments, the desired time may be the time taken for the laser to reach a steady state, in which case the solution is the steady-state solution.

[0114] Structural modifications to an optical computing system as described herein will be understood as being applicable to a method using such an optical computing system. For example the method may include splitting beams, connecting 4f telescopes, including additional optical cavities and the like.

[0115] In order to implement the optical computing system and the method according to embodiments of the present invention, a computer processor may receive instructions and data from a memory and communicate digital signals with the optical computing system, as known in the art.

[0116] Reference is now made to Fig. 5, which is a block diagram of an exemplary computing device 500 which may be used with embodiments of the present invention. Computing device 500 may perform steps such as associating between mathematical terms of the PDE and physical characteristics of the laser beam.

[0117] Computing device 500 may include a controller or processor 505 that may be, for example, a central processing unit processor (CPU), a chip or any suitable computing or computational device, an operating system 515, a memory 520, a storage 530, input devices 535 and output devices 540.

[0118] Operating system 515 may be or may include any code segment designed and / or configured to perform tasks involving coordination, scheduling, arbitration, supervising, controlling or otherwise managing operation of computing device 500, for example, scheduling execution of programs. Memory 520 may be or may include, for example, a Random Access Memory (RAM), a read only memory (ROM), a Dynamic RAM (DRAM), a Synchronous DRAM (SD-RAM), a double data rate (DDR) memory chip, a Flash memory, a volatile memory, a non-volatile memory, a cache memory, a buffer, a short term memory unit, a long term memory unit, or other suitable memory units or storageP-641241 -PCunits. Memory 520 may be or may include a plurality of, possibly different, memory units. Memory 520 may store for example, instructions to carry out a method (e.g., code 525), such as method 300, and / or data such as any of the equations described herein, for example governing laser rate equations (4) - (6).

[0119] Executable code 525 may be any executable code, e.g., an application, a program, a process, task or script. Executable code 525 may be executed by controller 505 possibly under control of operating system 515. In some embodiments, more than one computing device 500 or components of device 500 may be used for multiple functions described herein. For the various modules and functions described herein, one or more computing devices 500 or components of computing device 500 may be used. Devices that include components similar or different to those included in computing device 500 may be used, and may be connected to a network and used as a system. One or more processor(s) 505 may be configured to carry out embodiments of the present invention by for example executing software or code. Storage 530 may be or may include, for example, a hard disk drive, a floppy disk drive, a Compact Disk (CD) drive, a CD-Recordable (CD-R) drive, a universal serial bus (USB) device or other suitable removable and / or fixed storage unit. Storage 530 may include cloudbased storage. Storage 530 may include database storage. In some embodiments, some of the components shown in Fig. 5 may be omitted.

[0120] Input devices 535 may be or may include a mouse, a keyboard, a touch screen or pad or any suitable input device. It will be recognized that any suitable number of input devices may be operatively connected to computing device 500 as shown by block 535. Output devices 540 may include one or more displays, speakers and / or any other suitable output devices. It will be recognized that any suitable number of output devices may be operatively connected to computing device 500 as shown by block 540. Any applicable input / output (I / O) devices may be connected to computing device 500, for example, a wired or wireless network interface card (NIC), a modem, printer or facsimile machine, a universal serial bus (USB) device or external hard drive may be included in input devices 535 and / or output devices 540.

[0121] Embodiments of the invention may include one or more article(s) (e.g., memory 520 or storage 530) such as a computer or processor non-transitory readable medium, or a computer or processor non-transitory storage medium, such as for example a memory, a disk drive, or a USB flash memory, encoding, including or storing instructions, e.g., computer-executable instructions, which, when executed by a processor or controller, carry out methods disclosed herein.

[0122] As will be appreciated by one skilled in the art, aspects of the present invention may be embodied as a system, method and / or computer program product. A computer program product mayP-641241 -PCcontain code which causes an optical computing system, such as an optical computing system connected to a (“traditional” or non-optical) computing device / processor to perform method steps as described herein. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit”, “module” or “system.” Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied thereon.

[0123] Any combination of one or more computer readable medium(s) may be utilized. The computer readable medium may be a computer readable signal medium or a computer readable storage medium. A computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of the computer readable storage medium would include the following: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fibre, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium may be any tangible medium that can contain or store a program for use by or in connection with an instruction execution system, apparatus, or device.

[0124] Program code embodied on a computer readable medium may be transmitted using any appropriate medium, including but not limited to wireless, wireline, optical fibre cable, RF, etc., or any suitable combination of the foregoing.

[0125] Computer program code for carrying out operations for aspects of the present invention may be written in any combination of one or more programming languages, including an object-oriented programming language such as Java, Smalltalk, JavaScript Object Notation (JSON), C++ or the like and conventional procedural programming languages, such as the “C” programming language or similar programming languages. The program code may execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider).P-641241 -PC

[0126] Aspects of the present invention are described above with reference to flowchart illustrations and / or portion diagrams of methods, apparatus (systems) and computer program products according to embodiments of the invention. It will be understood that each portion of the flowchart illustrations and / or portion diagrams, and combinations of portions in the flowchart illustrations and / or portion diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general -purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart and / or portion diagram portion or portions.

[0127] These computer program instructions may also be stored in a computer readable medium that can direct a computer, other programmable data processing apparatus, or other devices to function in a particular manner, such that the instructions stored in the computer readable medium produce an article of manufacture including instructions which implement the function / act specified in the flowchart and / or portion diagram portion or portions.

[0128] The computer program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other devices to cause a series of operational steps to be performed on the computer, other programmable apparatus or other devices to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions / acts specified in the flowchart and / or portion diagram portion or portions.

[0129] The aforementioned flowchart and diagrams illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present invention. In this regard, each portion in the flowchart or portion diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the portion may occur out of the order noted in the figures. For example, two portions shown in succession may, in fact, be executed substantially concurrently, or the portions may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each portion of the portion diagrams and / or flowchart illustration, and combinations of portions in the portion diagrams and / or flowchart illustration, can be implemented by special purpose hardware-based systems that performP-641241 -PCthe specified functions or acts, or combinations of special purpose hardware and computer instructions.

[0130] In the above description, an embodiment is an example or implementation of the inventions. The various appearances of “one embodiment”, “an embodiment”, or "some embodiments" do not necessarily all refer to the same embodiments.

[0131] Although various features of the invention may be described in the context of a single embodiment, the features may also be provided separately or in any suitable combination. Conversely, although the invention may be described herein in the context of separate embodiments for clarity, the invention may also be implemented in a single embodiment.

[0132] Reference in the specification to “some embodiments”, “an embodiment”, “one embodiment” or “other embodiments” means that a particular feature, structure, or characteristic described in connection with the embodiments is included in at least some embodiments, but not necessarily all embodiments, of the inventions.

[0133] It is to be understood that the phraseology and terminology employed herein is not to be construed as limiting and are for descriptive purpose only.

[0134] The principles and uses of the teachings of the present invention may be better understood with reference to the accompanying description, figures and examples.

[0135] It is to be understood that the details set forth herein do not construe a limitation to an application of the invention.

[0136] Furthermore, it is to be understood that the invention can be carried out or practiced in various ways and that the invention can be implemented in embodiments other than the ones outlined in the description above.

[0137] It is to be understood that the terms “including”, “comprising”, “consisting of’ and grammatical variants thereof do not preclude the addition of one or more components, features, steps, or integers or groups thereof and that the terms are to be construed as specifying components, features, steps, or integers.

[0138] If the specification or claims refer to “an additional” element, that does not preclude there being more than one of the additional elements.

[0139] It is to be understood that where the claims or specification refer to “a” or “an” element, such reference is not construed that there is only one of that elements.

[0140] It is to be understood that where the specification states that a component, feature, structure, or characteristic “may”, “might”, “can” or “could” be included, that particular component, feature, structure, or characteristic is not required to be included.P-641241 -PC

[0141] Where applicable, although state diagrams, flow diagrams or both may be used to describe embodiments, the invention is not limited to those diagrams or to the corresponding descriptions. For example, flow need not move through each illustrated box or state, or in exactly the same order as illustrated and described.

[0142] Methods of the present invention may be implemented by performing or completing manually, automatically, or a combination thereof, selected steps or tasks.

[0143] The term “method” may refer to manners, means, techniques and procedures for accomplishing a given task including, but not limited to, those manners, means, techniques and procedures either known to, or readily developed from known manners, means, techniques and procedures by practitioners of the art to which the invention belongs.

[0144] The descriptions, examples, methods and materials presented in the claims and the specification are not to be construed as limiting but rather as illustrative only.

[0145] Meanings of technical and scientific terms used herein are to be commonly understood as by one of ordinary skill in the art to which the invention belongs, unless otherwise defined.

[0146] The present invention may be implemented in the testing or practice with methods and materials equivalent or similar to those described herein.

[0147] Any publications, including patents, patent applications and articles, referenced or mentioned in this specification are herein incorporated in their entirety into the specification, to the same extent as if each individual publication was specifically and individually indicated to be incorporated herein. In addition, citation or identification of any reference in the description of some embodiments of the invention shall not be construed as an admission that such reference is available as prior art to the present invention.

[0148] While the invention has been described with respect to a limited number of embodiments, these should not be construed as limitations on the scope of the invention, but rather as exemplifications of some of the preferred embodiments. Other possible variations, modifications, and applications are also within the scope of the invention. Accordingly, the scope of the invention should not be limited by what has thus far been described, but by the appended claims and their legal equivalents.

Claims

P-641241 -PCCLAIMS1. An optical computing system for solving partial differential equations (PDEs), comprising:a degenerate cavity laser (DCL) comprising an imaging system located within an optical cavity of the DCL, the DCL configured to generate a first laser beam which iteratively performs a number of round trip propagations within the optical cavity;one or more optical elements, located at one or more optical planes of the imaging system, configured to change at least one physical characteristic of the first laser beam, the one or more optical elements selectively configured to encode one or more mathematical operators of the PDE;at least one computer processor; anda memory containing instructions which, when executed by the at least one computer processor, cause the at least one computer processor to:receive a PDE to be solved;perform a first association associating a spatial coordinate grid with an image plane of the imaging system;perform a second association associating a time coordinate with a number of round trip propagations of the first laser beam; andoutput a solution of the PDE at a desired time by measuring the electric field of the first laser beam after a number of round trip propagations corresponding to the desired time, wherein a complex amplitude and phase of the measured electric field is converted to the solution based on the first and second associations.

2. The system of claim 1, further comprising a unidirectional coupled laser, wherein boundary conditions are enforced by the unidirectional coupling.

3. The system of claim 1, wherein boundary conditions are enforced by controlling one of the one or more optical elements to inject a second laser beam with known physical characteristics at a position in the image plane associated with a boundary condition.

4. The system of claim 1, wherein:the one or more optical elements are chosen from the list including: diffractive optical elements; spatial light modulators; beam splitters; beam combiners; lenses; mirrors; gain media; wave plates; digital micromirror devices; and / or nonlinear optical materials; andP-641241 -PCthe physical characteristics of a laser beam include at least one of: phase; amplitude; polarization; spatial profile; direction of propagation; spectral content; and number of split beams.

5. The system of claim 1, wherein the imaging system comprises one or multiple 4f telescopes, wherein the multiple 4f telescopes are connected in series or in parallel.

6. The system of claim 1, wherein an optical element of the one or more optical elements is configured to split the first laser beam into a number of laser beams corresponding to a number of mathematical terms of the PDE.

7. The system of claim 1, wherein an optical element located at an optical plane is configured to change the phase or amplitude of a laser beam, the change corresponding to a scalar factor in the PDE.

8. The system of claim 1, wherein at least some of the one or more optical elements are located at consecutive planes in the optical cavity, corresponding to a composition of functions in the PDE.

9. The system of claim 1, wherein the one or more optical elements comprises a nonlinear optical material, corresponding to a nonlinear operation in the PDE.

10. The system of claim 1, further comprising a number of optical cavities corresponding to the number of dependent variables of the PDE, wherein the optical cavities are connected according to the PDE.

11. The system according to any preceding claim, wherein coupled lasers are configured to represent a spatial variable of the PDE,12. The system according to any preceding claim, wherein a split laser beam is configured to represent a spatial operator of the PDE.

13. The system of claim 1, further comprising a camera configured to capture interference patterns between spots of the same laser beam.P-641241 -PC14. The system of claim 1, wherein the desired time is the time taken for the system to reach a steady state and the solution is the steady-state solution.

15. A method of optically computing solutions of partial differential equations (PDEs), the method comprising:receiving a PDE to be solved;generating a first laser beam which iteratively performs a number of round trip propagations within an optical cavity of a degenerate cavity laser (DCL);changing at least one physical characteristic of the first laser beam using one or more optical elements located at one or more optical planes of an imaging system located within the optical cavity;performing a first association associating a spatial coordinate grid with an image plane of the imaging system;performing a second association associating a time coordinate with a number of round trip propagations of the first laser beam; andoutputting a solution of the PDE at a desired time by measuring the electric field of the first laser beam after a number of round trip propagations corresponding to the desired time, wherein a complex amplitude and phase of the measured electric field is converted to the solution based on the first and second association,wherein the one or more optical elements are selectively configured to encode one or more mathematical operators of the PDE.

16. The method of claim 15, further comprising enforcing boundary conditions using unidirectional coupling.

17. The method of claim 15, further comprising enforcing boundary conditions using one of the one or more optical elements to inject a second laser beam with known physical characteristics at a position in the image plane associated with a boundary condition.

18. The method of claim 15, wherein:the one or more optical elements are chosen from the list comprising: diffractive optical elements; spatial light modulators; beam splitters; beam combiners; lenses; mirrors; gain media; wave plates; digital micromirror devices; and / or nonlinear optical materials; andP-641241 -PCthe physical characteristics of a laser beam include at least one of: phase; amplitude; polarization; spatial profile; direction of propagation; spectral content; and number of split beams.

19. The method of claim 15, further comprising adding to the imaging system one or multiple 4f telescopes, wherein the multiple 4f telescope are connected in series or in parallel.

20. The method of claim 15, further comprising using an optical element of the one or more optical elements to split the first laser beam into a number of laser beams corresponding to a number of mathematical terms of the PDE.

Images