704 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial Note that,in the receiving analog crossbar circuit,the phases are set to minus the corresponding phases in the receiving vectors themselves.This is because,in summing the currents in this crossbar,we are actually performing the inner pro- duct ()where is the column vector of received voltages [Vgn Vgin2 Vgin3],so as to extract the appropriate component of the signal. Since the中rg）vector is in "bra”form(φgin in the inner product,we must take the complex conjugate of the phase factors. Hence in this way,starting out with three quite separate signals,we are able to transmit them through the system and recover the original signals again.Note that the three channels here have no crosstalk even though the waves from each source are mixed at the three receiving points.This remains true even if the three"channels"in the system have different coupling strengths (as given by the singular values s,),as they do here, with relative power strengths of 3.41,2.89,and 1.37,respectively.Taken together, these three channels use up all the available power coupling strength,as given by the sum rule [Eq.(47)or,more generally,Eq.(36)]. The mathematical function performed by each of the crossbar circuits is,of course,sim- ply a (complex)matrix multiplication.By use of appropriate analog-to-digital and digital-to-analog conversion,such multiplications could be performed digitally instead. 4.2b.Optical Systems At optical frequencies,we generally cannot measure the field directly (certainly not in real time),and circuit approaches as in Fig.4 are not viable.Indeed,until recently,it was not generally understood in optics how to separate overlapping optical signals(without fundamental loss)to turn them into individual output beams,which is a process we require at the receiver in our scheme.Similarly,it was not clear how to losslessly gen- erate arbitrary linear superpositions of inputs to give overlapping outputs,as required at the source.Indeed,without some apparatus to perform such functions,in optics the multiple-channel schemes presented here would remain mathematical curiosities. Recently,however,specific schemes have been devised for both creating and sepa- rating arbitrary linear combinations of overlapping optical beams,and for emulating arbitrary linear optical components [24-27].Indeed,these schemes constitute the first proof that arbitrary linear optics is possible [25];the proof is entirely constructive because it shows specifically how to make the optical system,at least in principle. These schemes rely on meshes of two-beam interferometers,in appropriate architec- tures and with associated algorithms [24-28]to allow the meshes to be set up.We will not review these in detail in this article,but some key aspects are important here.In some of the architectures,the setup of the mesh (and,if required,the calculation of the necessary settings of the interferometers)can be entirely progressive [24-28].The setup of the mesh can also be accomplished by "training"the mesh with the beams of interest [24-28],based on a sequence of single-parameter power minimizations, without calculation or calibration. For our present example,the electronic matrix multiplications implicit in the analog crossbar networks at the source and at the receiver can be implemented instead using the "triangular"source mesh and receiving mesh,respectively,as in Fig.5(a).These meshes,which can also be viewed as analog crossbars,work directly by interfering beams in waveguides and waveguide couplers.The light in these meshes flows through them without fundamental loss,and they mathematically represent"unitary" (lossless)matrix multiplications. In Fig.5,we imagine that the input signals,instead of being voltages as in Fig.4,are the amplitudes Esin,Esi2,and Esin3 of the waves in single-(propagating)-mode input
Note that, in the receiving analog crossbar circuit, the phases are set to minus the corresponding phases in the receiving vectors themselves. This is because, in summing the currents in this crossbar, we are actually performing the inner product hϕRqjVRIni, where jVRIni is the column vector of received voltages VRIn1 VRIn2 VRIn3 T , so as to extract the appropriate component of the signal. Since the jϕRqi vector is in “bra” form hϕRInj in the inner product, we must take the complex conjugate of the phase factors. Hence in this way, starting out with three quite separate signals, we are able to transmit them through the system and recover the original signals again. Note that the three channels here have no crosstalk even though the waves from each source are mixed at the three receiving points. This remains true even if the three “channels” in the system have different coupling strengths (as given by the singular values sj), as they do here, with relative power strengths of 3.41, 2.89, and 1.37, respectively. Taken together, these three channels use up all the available power coupling strength, as given by the sum rule [Eq. (47) or, more generally, Eq. (36)]. The mathematical function performed by each of the crossbar circuits is, of course, simply a (complex) matrix multiplication. By use of appropriate analog-to-digital and digital-to-analog conversion, such multiplications could be performed digitally instead. 4.2b. Optical Systems At optical frequencies, we generally cannot measure the field directly (certainly not in real time), and circuit approaches as in Fig. 4 are not viable. Indeed, until recently, it was not generally understood in optics how to separate overlapping optical signals (without fundamental loss) to turn them into individual output beams, which is a process we require at the receiver in our scheme. Similarly, it was not clear how to losslessly generate arbitrary linear superpositions of inputs to give overlapping outputs, as required at the source. Indeed, without some apparatus to perform such functions, in optics the multiple-channel schemes presented here would remain mathematical curiosities. Recently, however, specific schemes have been devised for both creating and separating arbitrary linear combinations of overlapping optical beams, and for emulating arbitrary linear optical components [24–27]. Indeed, these schemes constitute the first proof that arbitrary linear optics is possible [25]; the proof is entirely constructive because it shows specifically how to make the optical system, at least in principle. These schemes rely on meshes of two-beam interferometers, in appropriate architectures and with associated algorithms [24–28] to allow the meshes to be set up. We will not review these in detail in this article, but some key aspects are important here. In some of the architectures, the setup of the mesh (and, if required, the calculation of the necessary settings of the interferometers) can be entirely progressive [24–28]. The setup of the mesh can also be accomplished by “training” the mesh with the beams of interest [24–28], based on a sequence of single-parameter power minimizations, without calculation or calibration. For our present example, the electronic matrix multiplications implicit in the analog crossbar networks at the source and at the receiver can be implemented instead using the “triangular” source mesh and receiving mesh, respectively, as in Fig. 5(a). These meshes, which can also be viewed as analog crossbars, work directly by interfering beams in waveguides and waveguide couplers. The light in these meshes flows through them without fundamental loss, and they mathematically represent “unitary” (lossless) matrix multiplications. In Fig. 5, we imagine that the input signals, instead of being voltages as in Fig. 4, are the amplitudes ESIn1, ESIn2, and ESIn3 of the waves in single-(propagating)-mode input 704 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial
Tutorial Vol.11,No.3/September 2019/Advances in Optics and Photonics 705 waveguides.By setting up the phase shifters and interferometers appropriately,the necessary superpositions are created as the amplitudes Es1,Es2,and Es3 in the output waveguides.These amplitudes then feed the point sources at the corresponding posi- tions rsi,rs2,and rs3. In this example,we pretend that the outputs of those guides essentially represent point sources of waves that we can also approximate as being scalar(which is reasonable if we consider only one polarization for the moment).At the receiving end,we imagine that the inputs to receiving mesh waveguides,at points r,rg2,and r3,are essentially "point"receiving elements to couple light into these waveguides,with corresponding waveguide amplitudes ERi,Eg2,and ER3,respectively.The receiving mesh then per- forms the appropriate matrix multiplication to separate out the signals(again without fundamental loss)to the output amplitudes Egout,Egout2,and Egout3,recreating the input bit streams. In Fig.5(b),we show one example way of making the necessary 2 x 2 interferometer block.This block should be able to control the"split ratio"of the interferometer-how the input power in,say,the top-left waveguide is split between the two output wave- guides on the right;this can be accomplished by controlling the phase angle 0 (achieved by differentially driving the two phase shifters on the arms of the interfer- ometer).The block should also independently be able to set one other phase on the outputs;in this example,this is achieved by the "common mode"drive of the phase shifters in the block,setting phase angle [140]. The setup of these unitary meshes is relatively straightforward [141].Such a mesh has exactly the right number of independent real parameters (here,nine phase shifters altogether)to construct an arbitrary 3 x 3 unitary matrix.Such design calculations are presented explicitly in [25]together with the self-configuring algorithms to allow direct training. In a real system,rather than"bare"waveguide outputs and inputs for communications, likely one would add some optics,such as collimating lenses,in front of the wave- guides,to avoid sending power in unnecessary directions.However,for our tutorial purposes at the moment,we will omit such optics (though it can ultimately also be handled by this approach by including the optics in the Green's function for the system). Figure 5 (a) Source mesh Receiving mesh 1 Ercm nrL s11 n Er nnn S22 31 R22 R31 (b) 5+日 Waveguide 6-8/2 ☐ 2×2 Phase shifter interferometer Example Mach-Zehnder 2x2 and phase shifter interferometer and phase shifter (a)Interferometer mesh architectures to generate and superpose the necessary source communications mode source vectors from the separate input signals on the left and to separate out the corresponding communications mode receiving vectors on the right to reconstruct the original channels of information.These processes work directly by interfering beams and without fundamental loss in the meshes.(b)Key to the various elements in (a).One example form of Mach-Zehnder interferometer and phase shifter is shown that has the necessary functions for the 2 x 2 blocks
waveguides. By setting up the phase shifters and interferometers appropriately, the necessary superpositions are created as the amplitudes ES1, ES2, and ES3 in the output waveguides. These amplitudes then feed the point sources at the corresponding positions rS1, rS2, and rS3. In this example, we pretend that the outputs of those guides essentially represent point sources of waves that we can also approximate as being scalar (which is reasonable if we consider only one polarization for the moment). At the receiving end, we imagine that the inputs to receiving mesh waveguides, at points rR1, rR2, and rR3, are essentially “point” receiving elements to couple light into these waveguides, with corresponding waveguide amplitudes ER1, ER2, and ER3, respectively. The receiving mesh then performs the appropriate matrix multiplication to separate out the signals (again without fundamental loss) to the output amplitudes EROut1, EROut2, and EROut3, recreating the input bit streams. In Fig. 5(b), we show one example way of making the necessary 2 × 2 interferometer block. This block should be able to control the “split ratio” of the interferometer—how the input power in, say, the top-left waveguide is split between the two output waveguides on the right; this can be accomplished by controlling the phase angle θ (achieved by differentially driving the two phase shifters on the arms of the interferometer). The block should also independently be able to set one other phase on the outputs; in this example, this is achieved by the “common mode” drive of the phase shifters in the block, setting phase angle ϕ [140]. The setup of these unitary meshes is relatively straightforward [141]. Such a mesh has exactly the right number of independent real parameters (here, nine phase shifters altogether) to construct an arbitrary 3 × 3 unitary matrix. Such design calculations are presented explicitly in [25] together with the self-configuring algorithms to allow direct training. In a real system, rather than “bare” waveguide outputs and inputs for communications, likely one would add some optics, such as collimating lenses, in front of the waveguides, to avoid sending power in unnecessary directions. However, for our tutorial purposes at the moment, we will omit such optics (though it can ultimately also be handled by this approach by including the optics in the Green’s function for the system). Figure 5 (a) Interferometer mesh architectures to generate and superpose the necessary source communications mode source vectors from the separate input signals on the left and to separate out the corresponding communications mode receiving vectors on the right to reconstruct the original channels of information. These processes work directly by interfering beams and without fundamental loss in the meshes. (b) Key to the various elements in (a). One example form of Mach–Zehnder interferometer and phase shifter is shown that has the necessary functions for the 2 × 2 blocks. Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 705
706 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial The kind of architecture shown in Fig.5(a),with unitary meshes at both sides,has one other interesting property worth noting here.There is an iterative algorithm [12], based only on overall power maximization on one channel at a time,and working forwards and backwards through the entire system,that allows this system itself to find the best coupled channels.Essentially,by running such an algorithm,this sys- tem physically can find the SVD of the coupling operator Gsk between the source points and the receiver points,without calculations.The results are then effectively stored as the settings of the various elements in the two meshes.This algorithm also still works even if there are other optics or scatterers between the source and receiver points,and so gives a way of finding the best orthogonal channels through any fixed linear optical system at a given frequency. 4.2c.Larger Systems As we consider larger numbers of source and receiver points,the specific approaches in Figs.4 and 5 would face technological limits of various kinds,especially for the purely optical approach of Fig.5.However,practically,the ability to make large num- bers of interferometers has been developing rapidly;working systems with up to hun- dreds of interferometers [109-116]and small self-configuring systems [27,29,117] have both been demonstrated recently.Extensions to thousands of interferometers may be feasible with current technology,and that number may not represent any par- ticular fundamental limit.Indeed,these demonstrations and the potential for expand- ing to larger systems are reasons why we consider these"modal"approaches.We need the modal approaches both in wireless systems,where they can be viewed as exten- sions to MIMO(multiple-input multiple-output)antenna and communications sys- tems,and in optical systems,where we are now able to explore such configurable and optimizable multiple-channel systems. Whether we choose to make systems as in Fig.4 or Fig.5,these kinds of systems show the upper limits in terms of orthogonal channels and coupling strengths of what could be achieved through such linear processing and the resulting optimum channels. The scheme of Fig.5 gives a method in principle of constructing arbitrary unitary(and hence nominally lossless)transforms of a given number of inputs and outputs(given ideal physical components).It operates with the minimum number of adjustable com- ponents and no loss in principle.No physical linear system can in principle do better than this scheme in constructing the communications modes for given numbers of source and receiver points.The existence of these approaches shows in principle that such systems could be made,both as actual physical systems up to some scale and as thought experiments at arbitrary scales for more basic discussions. In what follows,we look at a variety of systems with larger numbers of source and receiver points in various different geometries.This progressively introduces many different behaviors.Some of these behaviors at large scales can relate to those seen in conventional optical systems,but some quite general behaviors have no particular well-known precedents.Though there are just a few results that can be expressed in analytic approximations for specific classes of systems(e.g.,paraxial optics),there are several broader classes of behaviors that can be understood intuitively from these numerical simulations and some approximate heuristic results.These provide novel insights into communicating and sensing with waves in a wide range of systems,from acoustics,through radio and microwaves,to optics. 5.SCALAR WAVE EXAMPLES WITH POINT SOURCES AND RECEIVERS Now we continue to larger numbers and other geometries of source and receiver points to illustrate various behaviors
The kind of architecture shown in Fig. 5(a), with unitary meshes at both sides, has one other interesting property worth noting here. There is an iterative algorithm [12], based only on overall power maximization on one channel at a time, and working forwards and backwards through the entire system, that allows this system itself to find the best coupled channels. Essentially, by running such an algorithm, this system physically can find the SVD of the coupling operator GSR between the source points and the receiver points, without calculations. The results are then effectively stored as the settings of the various elements in the two meshes. This algorithm also still works even if there are other optics or scatterers between the source and receiver points, and so gives a way of finding the best orthogonal channels through any fixed linear optical system at a given frequency. 4.2c. Larger Systems As we consider larger numbers of source and receiver points, the specific approaches in Figs. 4 and 5 would face technological limits of various kinds, especially for the purely optical approach of Fig. 5. However, practically, the ability to make large numbers of interferometers has been developing rapidly; working systems with up to hundreds of interferometers [109–116] and small self-configuring systems [27,29,117] have both been demonstrated recently. Extensions to thousands of interferometers may be feasible with current technology, and that number may not represent any particular fundamental limit. Indeed, these demonstrations and the potential for expanding to larger systems are reasons why we consider these “modal” approaches. We need the modal approaches both in wireless systems, where they can be viewed as extensions to MIMO (multiple-input multiple-output) antenna and communications systems, and in optical systems, where we are now able to explore such configurable and optimizable multiple-channel systems. Whether we choose to make systems as in Fig. 4 or Fig. 5, these kinds of systems show the upper limits in terms of orthogonal channels and coupling strengths of what could be achieved through such linear processing and the resulting optimum channels. The scheme of Fig. 5 gives a method in principle of constructing arbitrary unitary (and hence nominally lossless) transforms of a given number of inputs and outputs (given ideal physical components). It operates with the minimum number of adjustable components and no loss in principle. No physical linear system can in principle do better than this scheme in constructing the communications modes for given numbers of source and receiver points. The existence of these approaches shows in principle that such systems could be made, both as actual physical systems up to some scale and as thought experiments at arbitrary scales for more basic discussions. In what follows, we look at a variety of systems with larger numbers of source and receiver points in various different geometries. This progressively introduces many different behaviors. Some of these behaviors at large scales can relate to those seen in conventional optical systems, but some quite general behaviors have no particular well-known precedents. Though there are just a few results that can be expressed in analytic approximations for specific classes of systems (e.g., paraxial optics), there are several broader classes of behaviors that can be understood intuitively from these numerical simulations and some approximate heuristic results. These provide novel insights into communicating and sensing with waves in a wide range of systems, from acoustics, through radio and microwaves, to optics. 5. SCALAR WAVE EXAMPLES WITH POINT SOURCES AND RECEIVERS Now we continue to larger numbers and other geometries of source and receiver points to illustrate various behaviors. 706 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial
Tutorial Vol.11,No.3/September 2019/Advances in Optics and Photonics 707 5.1.Nine Sources and Nine Receivers in Parallel Lines Now we space Ns =9 point sources over the same total length of 44 as in the three- source case above (Fig.3),and similarly for the Ng=9 receiving points,which means the points are spaced by A/2 in each case [142]. 5.1a.Channels and Coupling Strengths Using the same numerical approach as in the 3 x 3 case above,but now with a 9 x 9 matrix,there are nine orthogonal source vectors and nine corresponding orthogonal receiving vectors,and the sum rule S is S=72.65/g3R (51) The results for the coupling strengths are summarized in Table 1. Though there are formally nine orthogonal channels,there are only three strongly coupled channels of approximately the same coupling strength,one other channel about half as strong,one weak channel,one very weak channel,and three other extremely weak channels.Though we have nine sources and receivers,we certainly do not have nine practically usable channels.We see immediately that increasing the number of sources and/or receivers in given source and receiving volumes past a certain point does not increase the number of well-coupled channels. The inability to form further well-coupled channels is being enforced by the sum rule S,and could be viewed as an effective "diffraction"limit. 5.1b.Modes and Beams Once the eigenvectors ls)of amplitudes of the sources in each mode are calculated, it is straightforward to calculate the resulting wave or"beam"at any point r in space. Explicitly,with Ns point sources,and writing out the jth (column)eigenvector as lws〉≡[hyhy…hwg]r (52 (with the superscript T indicating the transpose,used here just to save space),the corresponding (complex)wave at point r is 10 )=- epr-rDh 53) Ir-rsg In Fig.6,we have plotted the resulting amplitudes and phases of the first three(strong- est)modes,together with the resulting waves or "beams."In Fig.6,(a)is mode 1, Table 1.Mode Coupling Strengths for Nine Point Sources and Receivers Mode Number,j Is,P/gix %of S Cum.of S 20.73 28.54 28.54 2039 28.07 56.61 19.09 26.28 82.89 10.41 14.34 97.23 1.90 2.62 99.84 0.11 0.16 ~100 0.0028 0.0038 ~100 0.000027 0.000037 ~100 0.000000065 0.000000089 ~100
5.1. Nine Sources and Nine Receivers in Parallel Lines Now we space NS 9 point sources over the same total length of 4λ as in the threesource case above (Fig. 3), and similarly for the NR 9 receiving points, which means the points are spaced by λ∕2 in each case [142]. 5.1a. Channels and Coupling Strengths Using the same numerical approach as in the 3 × 3 case above, but now with a 9 × 9 matrix, there are nine orthogonal source vectors and nine corresponding orthogonal receiving vectors, and the sum rule S is S 72.65∕g2 SR: (51) The results for the coupling strengths are summarized in Table 1. Though there are formally nine orthogonal channels, there are only three strongly coupled channels of approximately the same coupling strength, one other channel about half as strong, one weak channel, one very weak channel, and three other extremely weak channels. Though we have nine sources and receivers, we certainly do not have nine practically usable channels. We see immediately that increasing the number of sources and/or receivers in given source and receiving volumes past a certain point does not increase the number of well-coupled channels. The inability to form further well-coupled channels is being enforced by the sum rule S, and could be viewed as an effective “diffraction” limit. 5.1b. Modes and Beams Once the eigenvectors jψSji of amplitudes of the sources in each mode are calculated, it is straightforward to calculate the resulting wave or “beam” at any point r in space. Explicitly, with NS point sources, and writing out the jth (column) eigenvector as jψSji ≡ h1j h2j hNS j T (52) (with the superscript T indicating the transpose, used here just to save space), the corresponding (complex) wave at point r is ϕjr − 1 4π X NS q1 expikjr − rSqj jr − rSqj hqj: (53) In Fig. 6, we have plotted the resulting amplitudes and phases of the first three (strongest) modes, together with the resulting waves or “beams.” In Fig. 6, (a) is mode 1, Table 1. Mode Coupling Strengths for Nine Point Sources and Receivers Mode Number, j jsjj 2∕g2 SR % of S Cum. % of S 1 20.73 28.54 28.54 2 20.39 28.07 56.61 3 19.09 26.28 82.89 4 10.41 14.34 97.23 5 1.90 2.62 99.84 6 0.11 0.16 ∼100 7 0.0028 0.0038 ∼100 8 0.000027 0.000037 ∼100 9 0.000000065 0.000000089 ∼100 Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 707
708 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial (b)is mode 2,and (c)is mode 3.The wave plotted is the real part of the wave am- plitude,so it is essentially a snapshot of the wave at an arbitrary time,here for the values in the plane of the source and receiver points (i.e.,the plane of the "paper"). In Fig.6,we have also chosen the phase of all the source modes (source and receiving) to be zero in the middle of each mode [143].Here also,because of the symmetry of this problem,the receiving vectors are just the complex conjugates of the transmitting vectors.We have deliberately used a scale to plot the phase of the source points so that Figure 6 Source Source amplitude phase Source Receiver (a) 0 0元2r positions positions 2 6 c)0 0 Plots of the source relative amplitude,the source relative phases,and the resulting waves for the three most strongly coupled modes,(a),(b),and (c),respectively, for the nine point sources and receiving points shown.The phase of each source mode is chosen to be zero in the center of the mode.For graphic clarity,the wave is multi- plied by vz,where z is the horizontal position relative to the source plane;the actual wave decays in amplitude from left to right,and the real part of the wave is plotted in false color.To avoid singularities,the waves just next to the source are not shown,so the positions of the sources,as shown,are just outside the graphed region on the left. The source amplitudes of the points in the "source amplitude"and"source phase" plots are also indicated using an amplitude false color of the points
(b) is mode 2, and (c) is mode 3. The wave plotted is the real part of the wave amplitude, so it is essentially a snapshot of the wave at an arbitrary time, here for the values in the plane of the source and receiver points (i.e., the plane of the “paper”). In Fig. 6, we have also chosen the phase of all the source modes (source and receiving) to be zero in the middle of each mode [143]. Here also, because of the symmetry of this problem, the receiving vectors are just the complex conjugates of the transmitting vectors. We have deliberately used a scale to plot the phase of the source points so that Figure 6 Plots of the source relative amplitude, the source relative phases, and the resulting waves for the three most strongly coupled modes, (a), (b), and (c), respectively, for the nine point sources and receiving points shown. The phase of each source mode is chosen to be zero in the center of the mode. For graphic clarity, the wave is multiplied by ffiffiffiffiffi jzj p , where z is the horizontal position relative to the source plane; the actual wave decays in amplitude from left to right, and the real part of the wave is plotted in false color. To avoid singularities, the waves just next to the source are not shown, so the positions of the sources, as shown, are just outside the graphed region on the left. The source amplitudes of the points in the “source amplitude” and “source phase” plots are also indicated using an amplitude false color of the points. 708 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial