Maximum coupling between multimodel aperture antennas
Patent Information
- Application Number
- PCT/US2026/020529
- Authority / Receiving Office
- WO · WO
- Patent Type
- Applications
- Current Assignee / Owner
- Priority Date
- 2025-03-25
- Filing Date
- 2026-03-24
- Publication Date
- 2026-10-01
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Figure US2026020529_01102026_PF_FP_ABST
Abstract
Description
Attorney Docket No. 2115-008457-WO-POAMAXIMUM COUPLING BETWEEN MULTIMODEL APERTURE ANTENNASCROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claims the benefit of U.S. Provisional Application No.63 / 777,278, filed on March 25, 2025. The entire disclosure of the above application is incorporated herein by reference.FIELD
[0002] The present disclosure relates to determining maximum coupling between multimodal aperture antennas.BACKGROUND
[0003] With the maturation of metasurface technology, control of antenna aperture profiles has become possible with subwavelength precision. This capability of arbitrary aperture synthesis is expected to find application in not only far-field wireless links but also reactive and radiative near-field links for communication, power transfer, sensing, and even optical interconnects. This new-found capability has prompted this study into optimal apertures and their realization by leveraging advancements in commercial simulation.
[0004] In very early works, Borgiotti used results by Hu and Rumsey’s electromagnetic reaction concept to analytically consider coupling between planar apertures in the paraxial limit. He solved an eigenvalue problem of an integral operator to derive the apertures for maximum coupling. Borgiotti’s results were recently extended beyond the paraxial limit to design optimal apertures with practical metasurface antennas at mm-wave frequencies.
[0005] These works pursued an analytical approach, using the free-space Green’s function, to calculate the reaction integral between apertures. They considered the case of a continuous planar aperture and solved for the eigenfunctions of an infinite-dimensional integral operator. Key assumptions included the absence of interfering objects or scatterers and a free-space wireless link between two planar antennas. Additionally, low interaction between the apertures was assumed, with the transmitting aperture’s input impedance being independent of the receiving aperture.
[0006] A goal of this disclosure is to leverage metasurface technology to design maximally coupled aperture antennas in near-field applications. Thus, a moreAttorney Docket No. 2115-008457-WO-POAgeneralized design approach must be developed. Techniques are formulated to solve for field profiles that maximize power transfer between electrically large (many wavelengths) aperture antennas, even when the apertures are in the reactive or radiative near-field of each other, or in rich scattering environments with obstacles. If a multimode impedance matrix capturing the aperture interaction can be defined, the techniques presented here can maximize power transfer and the number of usable channels, under any of the aforementioned conditions. This approach even applies to dissimilar aperture antennas with differing numbers of modes.
[0007] This section provides background information related to the present disclosure which is not necessarily prior art.SUMMARY
[0008] This section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features.
[0009] A computer-implemented method is presented for determining channels that maximize electromagnetic coupling of power between a source antenna and a load antenna. As a starting point, the geometry for the source antenna and the geometry for the load antenna are provided. The source antenna is represented by N modes and the load antenna is represented by M modes, where N and M can differ.
[0010] A system impedance matrix representing the source antenna, the load antenna and the electromagnetic coupling between them is derived using the geometry for the source antenna and the geometry for the load antenna. In some embodiments, the system impedance matrix may also account for the electromagnetic environment between the antennas. In an example embodiment, the system impedance matrix is derived by simulating the source antenna and the load antenna using an electromagnetic field solver.
[0011] Next, an input impedance matrix is defined in terms of elements of the system impedance matrix and a load impedance matrix, and an output impedance matrix is defined in terms of elements of the system impedance matrix and a source impedance matrix. Matched termination impedance matrices are then calculated. That is, the source impedance matrix is impedance matched to the input impedance matrix, and the load impedance matrix is impedance matched to the output impedance matrix. Impedance matching includes applying the multimode maximum power transfer theorem for N-ports simultaneously at the ports of both the source antenna and the load antenna to find impedance matrices that maximize power. In the exampleAttorney Docket No. 2115-008457-WO-POAembodiment, the termination impedance matrices are calculated using a fixed-point iteration method although other techniques for computing fixed points is contemplated by this disclosure.
[0012] A transducer gain for power transfer between the source antenna and the load antenna is defined. More specifically, the transducer gain is expressed in terms of the source impedance matrix, the load impedance matrix and an excitation current for the source antenna, where the excitation current is expressed as a vector having an element for each of the N modes of the source antenna.
[0013] Lastly, the excitation current for the source antenna is determined, for example by optimizing the transducer gain. In the example embodiment, the excitation current is determined by expressing the transducer gain as a generalized Rayleigh quotient; applying a transformation to the generalized Rayleigh quotient to derive a standard Rayleigh quotient; computing eigenvalues and corresponding eigenvectors for the standard Rayleigh quotient; and transforming each eigenvector to a vector representing the excitation current that achieves the corresponding eigenvalue by inverting the transformation.
[0014] In another aspect, the multimodal matching approach can be extended to other types of devices (e.g., optical fibers, transmission lines, etc.) that can be characterized by a system impedance matrix. Thus, a computer-implemented method is also presented for determining a maximum electromagnetic coupling of power between a source device and a load device.
[0015] A system impedance matrix representing the source device and the load device and electromagnetic coupling between them using an electromagnetic field solver is first computed, where the source device exhibits N modes and the load device exhibits M modes. Next, a source impedance matrix is impedance matched to an input impedance matrix and a load impedance matrix is impedance matched to an output impedance matrix, where the input impedance matrix is defined in terms of elements of the system impedance matrix and the load impedance matrix and the output impedance matrix is defined in terms of elements of the system impedance matrix and the source impedance matrix.
[0016] A transducer gain for power transfer between the source device and the load device is then defined, where the transducer gain is expressed in terms of the system impedance matrix, source impedance matrix, the load impedance matrix and an input for the source device, where the input is expressed as a vector having anAttorney Docket No. 2115-008457-WO-POAelement for each of the N modes of the source device; and the input for the source device is determined by optimizing the transducer gain.
[0017] Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.BRIEF DESCRIPTION OF THE DRAWINGS
[0018] The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure.
[0019] Figure 1 is a diagram showing the wireless link between two multimodal antenna apertures; the transmitting aperture is excited by a vector of voltages which excite the multiple modes on each aperture.
[0020] Figure 2 is a diagram showing a general multimodal 2-port system modeled by an impedance matrix with source and load termination impedances Zs and ZL, and excited by a vector of voltages.
[0021] Figure 3 is a diagram showing an arbitrary multimodal 1 -port device with impedance matrix Z driven by a vector of voltages which excite the various modes.
[0022] Figure 4 is a flowchart depicts a technique for determining maximum electromagnetic coupling of power between a source antenna and a load antenna.
[0023] Figure 5 is a diagram of the kth eigenchannel coupling a pair of antennas, where the transmitter and receiver may support different modes.
[0024] Figure 6 is a flowchart showing a fixed-point iteration impedance matching algorithm.
[0025] Figure 7 is a diagram of two flared, circular apertures fed by overmolded waveguides of the same diameter.
[0026] Figure 8 is a diagram of an isolated, flared circular aperture fed by an overmolded waveguide of the same diameter.
[0027] Figure 9 are graphs showing coupling coefficients for the eigenchannels of a wireless link between two flared circular apertures having different separation distances.
[0028] Figure 10 are graphs showing full-wave verifications of the coupling coefficients for the eigenchannels of a wireless link between two flared circular apertures having different separation distances.Attorney Docket No. 2115-008457-WO-POA
[0029] Figure 11 are graphs showing optimal TM aperture fields (with a simultaneous match) for maximum power transfer between two apertures.
[0030] Figure 12 are graphs showing optimal TE aperture fields (with a simultaneous Hermitian match) for maximum power transfer between two apertures.
[0031] Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings.DETAILED DESCRIPTION
[0032] Example embodiments will now be described more fully with reference to the accompanying drawings.
[0033] An arbitrary pair of antennas can be modeled as a matrix operator. For the antenna coupling problem, it is convenient to use an impedance matrix, since this network representation is independent of the termination impedances. The aperture termination impedances serve as added degrees of freedom in design.
[0034] In some implementations, the antennas are aperture antennas having multiple physical ports. The physical ports are called modes and each port can be assigned complex voltage and current. Thus, a voltage vector for each aperture is a list of voltages at the physical ports and a current vector for each aperture is a list of currents at the physical ports.
[0035] In other implements, the antennas are aperture antennas having one port with multiple eigenmodes (i.e., waveguide modes). Each waveguide mode in each aperture can be assigned a complex weight on E and H independently. Thus, the voltage vector for each aperture is a list of weights on the modal E and the current vector for each aperture is a list of weights on the modal H. The mode fields are added together to get the total fields. The multimodal impedance matrix Z for two apertures can be found via full-wave simulation or using reaction integrals.
[0036] Let the transmitting or source aperture, represented by N modes, be denoted aperture 1 , and the receiving or load aperture, represented by M modes, be denoted aperture 2. In this disclosure, allow the modes of aperture 1 and aperture 2 to be different, where N need not be equal to M. Figure 1 shows the most general model of a multimodal network that characterizes the aperture interactions. The following matrix equation relates the modal voltages and currents.v = 7.1,where 7 e (1)V,l G <CN+M.Attorney Docket No. 2115-008457-WO-POAExpanded in block-matrix form with respect to the aperture ports, it becomes,v1,t1E V2J2GZuG CWxWZ22G CMXMZ12G CWxMZ21G CMxWThe vectors Vi and v2are the voltages of the modes on each aperture port. The vectors 4 and are the currents of the modes on each aperture port. The direction of is chosen out of the network as opposed to the typical definition, since this is a more natural choice when coupling power from aperture 1 to aperture 2. The definition of Z itself is unchanged from the typical one. The source and load impedances terminating the apertures are matrices, Zs and Zi_, as shown in Figures 2 and 3.
[0037] The voltage and current at port 1 are related to the source voltage "vs, the source impedance Zs, and the input impedance Zinat aperture 1 as follows,V1 —VS — ^inH (3)VS=(Zs + ...= = (4)where Zs, ZinG CWXWVSG CW.Similarly, the load voltage and current are related to the load impedance as follows,V2—ZLl2(5)where ZLG CMXM.
[0038] The source and load impedance matrices relate the voltage and current vectors at the aperture ports. If the impedance matrix of interest is diagonal, there is no modal coupling or mixing at the ports. However, generally these matrices can be non-diagonal.
[0039] Using (2) - (5), one can define the input impedance matrix at port 1 as,=Zu — Z12(zL+ Z22) Z21(6)Similarly, turning off the source on aperture 1 and driving aperture 2 with a test source yields the output impedance matrix at port 2,Attorney Docket No. 2115-008457-WO-POAZout—z22 z21(zs+Z11j z12,where ZoutG CMXM.This set of equations fully characterizes the multimodal coupled aperture system.
[0040] Multimodal power definitions and notations are introduced next.Consider a linear multimodal 1 -port device, with impedance matrix Z, driven by a multimodal source. Its current vector is defined into the device and the voltage vector appears across the device terminals as shown in Fig. 3.
[0041] The vector of complex modal input power is defined with the Hadamard product ° and the element-wise conjugate * as follows,PCop= |ro(Zi) (8)
[0042] Since the total power is the sum of individual modal powers, the total complex power can be written as a sesquilinear form using Z, with superscriptHdenoting the Hermitiantranspose as follows,pc= -tHv = - I2H7.I2 (9)The time-averaged power input to the multimodal device,91{PC} == |QIH7I + |(IH7I)*)— — H \-1 Z1 +-1 Z i I (10)2 2 I= -iHf-fz + z Yk2 \ 2 \ / / is a Hermitian form defined by the Hermitian part of Z, H{Z} =Z + ZI. Similarly, the reactive power input to the multimodal device,3{PC] = i3{FHZ?}(11)is a Hermitian form defined using the anti-Hermitian part of Z, / 1{Z} = Z - Z Therefore, one can generalize the impedances for the multimodal case as follows,Attorney Docket No. 2115-008457-WO-POAR = H{Z] = | + Z jX = |A{Z} = ^Z- Z ) (12)Z = R + jX.
[0043] Here, the resistance matrix, R, and the reactance matrix, X, are Hermitian matrices. They define Hermitian forms with the modal current that provide the total real and reactive modal input power.
[0044] Desoer extended the maximum power transfer theorem (MPTT) to multimodal 1 -port devices in 1973, via a maximizing current approach, which is stated here for completeness. Let the source impedance Zs- be fixed, and assume Rsis positive definite. For arbitrary but stationary modal voltage vs, the power dissipated in Z,51{PC} = |lHRl / _H -H\-1— - - -i (13)= — I Z^ + Z j R ( Z.s + Z J vsvaries only with Z . Under the restriction that R is also positive definite, (13) is maximized when,Z = (14)
[0045] Desoer noted that for a single source voltage vector, multiple impedance matrices existed that achieve his maximizing current. However, only the Hermitian-match, which is independent of vs, achieves the maximizing current for every possible vs.
[0046] This Hermitian-matched condition is the natural generalization of conjugate-matching in the scalar MPTT, and it gives the power available from the multimodal source by substituting (14) into (13),— -1Pavs=gVS ^SVS> (15)which agrees with the scalar case.
[0047] Figure 4 depicts a technique for determining maximum electromagnetic coupling of power between a source antenna and a load antenna. As a starting point, the geometry for the source antenna and the geometry for the load antenna are provided as indicated at 41. As noted above, the source antenna is represented by N modes and the load antenna is represented by M modes, where N and M can differ.Attorney Docket No. 2115-008457-WO-POA
[0048] A system impedance matrix representing the source antenna, the load antenna and the electromagnetic coupling between them is derived at 42 using the geometry for the source antenna and the geometry for the load antenna. In some embodiments, the system impedance matrix may also account for the electromagnetic environment between the antennas (e.g., obstacles, diffraction, scattering). In an example embodiment, the system impedance matrix is derived by simulating the source antenna and the load antenna using an electromagnetic field solver, such Ansys HFSS simulation software.
[0049] Next, an input impedance matrix is defined in terms of elements of the system impedance matrix and a load impedance matrix, and an output impedance matrix is defined in terms of elements of the system impedance matrix and a source impedance matrix as indicated at 43. Matched termination impedance matrices are then calculated at 44. That is, the source impedance matrix is impedance matched to the input impedance matrix, and the load impedance matrix is impedance matched to the output impedance matrix. Impedance matching includes applying the multimode maximum power transfer theorem for N-ports simultaneously at the ports of both the source antenna and the load antenna to find impedance matrices that maximize power. In the example embodiment, the termination impedance matrices are calculated using a fixed-point method as further described below.
[0050] A transducer gain for power transfer between the source antenna and the load antenna is defined at 46. More specifically, the transducer gain is expressed in terms of the source impedance matrix, the load impedance matrix and an excitation current for the source antenna, where the excitation current is expressed as a vector having an element for each of the N modes of the source antenna.
[0051] Lastly, the excitation current for the source antenna is determined at 47 by optimizing the transducer gain. In the example embodiment, the excitation current is determined by expressing the transducer gain as a generalized Rayleigh quotient; applying a transformation to the generalized Rayleigh quotient to derive a standard Rayleigh quotient; computing eigenvalues and corresponding eigenvectors for the standard Rayleigh quotient; and transforming each eigenvector to a vector representing the excitation current that achieves the corresponding eigenvalue by inverting the transformation, where the vector represents the excitation current. The computation for the excitation current is further described below.
[0052] In single-mode, two-port networks, maximum transducer gain is attained simply with a simultaneous conjugate impedance match at both ports.Attorney Docket No. 2115-008457-WO-POAHowever, in the case of multimode ports, both the termination impedances and the optimal weights of the modes must be solved for. Thus, multimodal transducer gain is defined before solving for simultaneous-matched termination impedance matrices. In addition, one can solve for the optimal weights of the modes. While the focus of this disclosure is to maximize coupling between apertures, the techniques described here are applicable to a more general class of multimodal 2-port devices. For generality, first derive the transducer gain matrix and the eigenchannels for arbitrary impedance matrix terminations. However, in the design process, impedance matrix matching is performed first. As a result, simplified expressions for the transducer gain matrix and the eigenchannels can be applied.
[0053] Here, the multimodal transducer gain is defined. The transducer gain is a worst case coupling metric for the apertures, depending both on Zs- and ZL. It is the most appropriate metric to quantify coupling performance. Note again that no assumptions are made about the aperture 1 and aperture 2 ports having the same number of modes.This power ratio, y, will simply be referred to as the coupling coefficient. Note that the power Hermitian forms can be defined by simple algebra in terms of any of the modal weight vectors at either port. However, if one considers aperture 1 as the transmitter, it is natural to define them in terms of a port-1 quantity, such as v or1.Here, one can use1.
[0054] First, consider the multimodal available source power. Substituting (4) into (15) givesPays ~ + Zin) R$ ^Zs- + Zin)l^. (1^)
[0055] Next, the multimodal power delivered to the receive / load aperture is given by,Wf} = {?"7tl2} = (18)Using (2) - (5), the currents at ports 1 and 2 can be related as,i2= M (19)where M = (zL+ Z22) Z21. (20)Thus, the power dissipated in the load is a function of the load impedance and the modal current vector at port 1.Attorney Docket No. 2115-008457-WO-POA= (21)
[0056] Now the coupling coefficient can be expressed as a generalized Rayleigh quotient,7( '2^ ) / =® r avs (22) / — H— =\4l^( M RLM Ji!(23)ll Rs fZs+Zin)1!This coupling coefficient, y, depends on three design variables.1) The termination impedance matrix of the multimodal source at the transmitting aperture, Zs-.2) The termination impedance matrix seen by the modes at the receiving aperture, the load impedance matrix ZL.3) The modal current vector at aperture 1 , i1;representing the complex weight of each mode on the transmitting aperture.These three design variables completely define the modal voltage and current at both ports. Applying waveguide circuit theory, the aperture fields are completely defined.
[0057] In maximizing the function y (Zs, ZL, ) over the design variables, note that the termination resistance matrices Rsand RLare restricted to be positive definite. In other words, for all non-zero modal currents, the terminations dissipate some power. This is reasonable because any practical source has ohmic loss, and the load impedance must be dissipative to extract real power from the system.
[0058] The independent maximum power transfer eigenchannels for a given pair of termination impedance matrices (Zs, ZL) is described next. It solves for optimal "i1 , one of the three design variables that affect the coupling coefficient y.
[0059] First, one must choose a transformation to rewrite y as a standard — -i Rayleigh quotient. Mathematically, this is accomplished by factorizing Rs. However, there is some physical intuition to be gleaned from the transform, which amounts to a resistance normalization.
[0060] Consider a general multimodal, one-port device with impedance Z and positive definite resistance R . Many factorizations of the following form exist,— — H—R = A A, (24)Attorney Docket No. 2115-008457-WO-POAwith invertible A. Each such factorization achieves a resistance normalization by defining a modified v* and i* from v and i as follows,— (—i* = At, v„ = I A I v (25)— f—H\~r= ^(Ai)HI A I v = ±iHv = Pc(26)Applying (25),v„ = I A j v = I A j Zi / — H\— 1-1= I A j ZA i,L / -I = - - 1(27)= (A ) (R+jX)A \ / - Z--i\H— 1\= I + j A j XA uone can define a modified impedance as follows,_ / _ -AZ* = I +j I A j XA (28)_ - - -iR = I,X„ = I A j XA . (29)Thus, the physical meaning of such transformations is that the resistance matrix with respect to the transformed voltage and current is the identity matrix. This also means that the time-averaged power is given by, 51{PC} = -||vj|2= -||ij|2, which generalizes the notion of normalizing a signal to a 1Q resistance.
[0061] Although there are many choices for A, two normalizing transforms are of particular interest. First, only one choice of A exists that is also Hermitian positive —1 / 2definite, denoted the principal square root R . It is given by the unitary diagonalization,— — H / — / Z— / — H\ —1 / 2— 1 / 2R = PAP = PJAP PJAP = R R (30)—1 / 2 = l^=HwhereR = P^AP . Note that the square root of a diagonal matrix is taken by applying the square root to the diagonal elements. Physically, the principal squareAttorney Docket No. 2115-008457-WO-POAroot transform normalizes the input resistance to the identity matrix and expresses the port voltage and current in terms of the original modes at the feed port.—1 / 2 — -1 / 2i = R t,v* = R v (31)
[0062] A second closely related transform instead puts the device eigenmodes to the fore.— ———H( I—~H\H( l—~H\R = PAP = JAP JAP (32)
[0063] Physically, this eigenmode transform normalizes the input resistance to the identity matrix. Specifically, it converts the port voltage and current to the— H eigenmodes of the resistance matrix of the device with the operator P , then normalizes by the eigenvalues of the resistance matrix, / Z— H |Z=H4= JAP I, v* = J A - IP v (33)
[0064] Armed with this physical intuition, one can return to the original problem of rewriting y as a traditional Rayleigh quotient. For simplicity, apply the principal square root to define the following transformation with respect to the source resistance matrix as in (31),—1 / 2 — -1 / 2li'=Ksli> vS' = Rs vs(34)
[0065] Physically, this transformation converts the source voltage vsand the port 1 current to the eigenmodes of the source resistance matrix with the operator — HPs, then normalizes by the corresponding source resistance eigenvalues, and finally — Hconverts back to the modes of the feed port with the operator Ps.
[0066] In the special case that the source is Hermitian-matched to the input of the multimodal 2-port, Rs= Rin. Thus, the source resistance eigenmodes become the same as the input resistance eigenmodes, and the input power at port 1 becomes equivalent to the available source power at port 1.
[0067] The normalization transform simplifies the available source power expression such that it only depends on the magnitude of vs>,Pays = ^RsVS= ^VS' = i || VS' ||2. (35)The aperture currents are related to the normalized source voltage by,— 1 / 2 - -VS' — K? (Zs + ^injll (36)Attorney Docket No. 2115-008457-WO-POAEqn. (36) can be applied to (23) to represent y as a traditional Rayleigh quotient._„ / = = xH=-1 / 2_-l / 2 / = = xll (^s+Zin) R.S Rs (Zs+Zin)ti (=H= =\ 4l^l M RLM II!'=-1 / 2 z= = . \H=~^(= = .Rs (^.S”*-ZjnJ I Rs ^Zs+Zfrijli ^s'— —1 / 2 / —H — H \1— H— — — . -1 — 1 / 2T=4RS ( Zs- + Z(nj M RLM ^ZS- + Zin) Rs(38)
[0068] From (37), it is well-known that the eigenvalues of T are stationary values of the coupling coefficient y with respect to vs>, given in decreasing order by, yW, (39)and the corresponding eigenvectors are stationary points of the coupling coefficient, given as the columns of,vs> i= ... v^) (40)
[0069] Thus, the largest eigenvalue y(1)maximizes the coupling coefficient for a pair of fixed termination impedances (ZsZL), and the corresponding eigenvector is the required normalized source voltage vector. The other eigenvectors, corresponding to large eigenvalues (coupling coefficients), are the next best orthogonal excitations. Since the eigenvectors are orthogonal by the spectral theorem and thus independent, the invertibility of (36) implies that the resulting vectors at port 1 ,T1 =^(zs+ Zin) Rs= (?« ... 7^) (41)are linearly independent and form a basis of excitation currents on port 1. Each of these current vectors excites an independent power transfer eigenchannel between the apertures.
[0070] Precisely, eachexcitation current on port 1 of aperture 1 couples power into port 2 of aperture 2 with a coupling coefficient given by the corresponding y(k)above. These independent excitations constitute multiple power coupling eigenchannels with eigenchannel strengths given by the eigenvalues. A diagram illustrating the ktheigenchannel is shown in Figure 5. This formulation and analysis is also applicable to power transfer problems through general multimodal devices.Attorney Docket No. 2115-008457-WO-POA
[0071] Since Zsand ZLare design variables, the maximum power transfer theorem for multimodal systems can be applied at both aperture ports simultaneously to find the impedance matrices that maximize power to the load,— — H — —H^S,m ~ Zjn^L,m ~ Zout , (42)where the subscript m designates that these terminations are Hermitian matched. This condition is the generalization of a simultaneous conjugate match applied to multimodal systems. Since the input impedance is a function of the load impedance and the output impedance is a function of the source impedance, the source and load impedances under a Hermitian match need to be solved simultaneously. By substituting (42) into (6) and (7), the equations for the matched impedances, Zs>7nand ZL m, become decoupled.^•S,m — W — H -1 —r1 =H(43) ^11—Z2i2^22 •S,m ^12 Z12^•L,m —H —H -1 —2R1 =H(44)r Z22—z12n 'L,m ^21 Z2IIt is evident that the right hand side of equation (43) depends only on Zs>7n. It is also evident that the right hand side of equation (44) depends only on ZL m. Thus, equations (43) and (44) represent dual fixed-point problems in Zs.mand ZL m, respectively. Only one of these problems must be solved to obtain the simultaneous Hermitian matched terminations, and this can be done iteratively using a fixed-point iteration algorithm. Lastly, under a Hermitian match, the following expression for the transducer gain matrix can be derived,— — -1 / 2— H — — — -1 / 2I'm — R.S'.m (45)with Mm= (zL m+ Z22) Z21(46)
[0072] Fixed-point iteration yields the simultaneous Hermitian matched termination impedances, Zs>7nand ZL m. This algorithm is a standard technique, and is particularly well known for finding fixed-points of functions over IK. In general, it can be applied to find fixed-points of functions over particular metric spaces (specifically, Banach, or complete normed vector spaces), which includes the space of complex matrices under any norm. For completeness, norms give a naturalAttorney Docket No. 2115-OQ8457-WO-POAdistance in the same way as the absolute value does within IK. Where the distance between two real numbers a and b is denoted by d(a, b) = |b - a|, the distance between two matrices A and B can be defined by d(A, B) = ||B - A|| for any matrix norm || ■ ||. Here, choose, somewhat arbitrarily, the distance metric induced by the Frobenius norm, defined for arbitrary impedance matrices by(47)which lets us consider the possible source and load impedance matrices as metric spaces. Consider the following non-linear mappings over these spaces, Ts<CNxNCNXNAND TL- CMXMCMXM.Ts {zs} =— H — H (48) Zll—Z2i2^22^ {zL} =— H — H=(49\-1= -i-H ) Z22—z122Rn Z12ZL + Z22) Z21Z21
[0073] Comparing (48) with (43) and (49) with (44), one can see that fixed-points of these mappings solve the simultaneous matching problem,Ts j Zsmf — Zs,m(50)TLLJj IZLLjfmlll.l I=^L Li ,m 11 L-
[0074] Clearly, solving either (43) or (44) for Zs>7nor ZL mimmediately provides the other, as both of the decoupled equations were defined directly from the simultaneous matching condition. Without loss of generality, Zs>7nis solved for here. The fixed-point iteration algorithm is illustrated in Figure 6. First, an appropriate seed matrix Zs>0is chosen (Typically Zs>0= 50K1)) and a sequence is constructedfrom the following recursion relationship with index variable p,%s,p+i=Ts{z.siPj (51 ){Zs,o> Zs... , Zs pr j (52)
[0075] This algorithm is considered converged to a fixed point if for a small prescribed e > 0, a p is found such that,d (js [z.s-.p j , Zs,p)p<e(53)which one can denote p'. This gives,Attorney Docket No. 2115-008457-WO-POAI'S.m. ~ ^S.p' ’ (54)— —H —H / —H —H \-1—H^•L,m=^22—Z12( ^S.p' T ^11 J Z21. (55)
[0076] Fixed-point iteration can be rigorously guaranteed to converge (uniquely) for functions that contract over some subspace on which they are closed. However, more generally fixed-point iteration may converge to attractor points, a concept more commonly studied in dynamical system theory. Convergence properties of the matching approach are beyond the scope of this disclosure. If the seed matrix Zs>0is chosen such that Rs>0is positive definite, the map Ts becomes closed to the subset of matrices with positive definite Hermitian part, i.e. all source impedances in the above sequence have Rs zpositive definite. Thus, if this sequence converges, it must converge to a source impedance with positive definite resistive part, which also implies by (42) that the corresponding converged load impedance has a positive definite resistive part. It is also straightforward to show that if the multimodal 2-port device is reciprocal, the algorithm will converge to terminations that are symmetric. Similar arguments can be made for the map TL.
[0077] Again, a more rigorous treatment of this simultaneous Hermitian matching algorithm is the subject of ongoing study. However, simple Monte Carlo analysis and simulation results suggest that the fixed point algorithm converges to unique dissipative terminations if R is strictly positive definite. In the well-conditioned case where the aperture system is passive and has at least a small amount of loss, this is always true.
[0078] Hereafter, converged Zs>7nand ZL mmatrices are assumed to have positive definite resistive parts. At very close distances with lossless apertures, R can have numerically zero eigenvalues in simulation. This may require diagonal loading of Z to obtain unique values of Zs>7nand ZL m, as shown in the appendix.
[0079] Next, a design example with continuous apertures is considered in simulation. A continuous aperture is implemented as a flared circular aperture fed by a circular waveguide of the same diameter, as shown in Figure 7. The waveguide fed aperture is perfect electric conducting (PEC) and the operating frequency is 30 GHz. The apertures are aligned, and only excited with azimuthally invariant modes. Thus, the solution is azimuthally invariant.Attorney Docket No. 2115-008457-WO-POA
[0080] By feeding the aperture with a 5A circular waveguide, one can ensure only the propagating azimuthal zeroth order modes are seen at the ports. This design reduces the dimensionality of the problem such that only need to consider the 9 propagating modes at each port plane (TM01, TM02, TM03, TM04, TM05, TE01, TE02, TE03, TE04). One can also reduce the length of the waveguide feeds and consider an additional set of accessible evanescent modes. The multimodal impedance matrix of this two-aperture system is found using the commercial Method of Moments (MoM) solver in Altair FEKO, where only the aforementioned propagating modes are considered at the waveguide ports. This simulation was repeated at various separation distances: 2.5A, 5A, 10A, 15A, 20A, 25A.
[0081] Once the interaction matrix Z is exported, one can implement the network theory approach described earlier in Python. First, the simultaneous Hermitian matched impedance matrices, Zs>7nand ZL mare computed by the fixed-point iteration algorithm with a convergence criterion of e = 10-5 and diagonal loading of 1 Q. Given these terminations along with Z, compute the transducer gain matrix Tm. Subsequently, compute the eigenvalues (coupling coefficients) and eigenvectors of the transducer gain matrix, and from this, the port-1 current excitation vectors for each eigenchannel. These current excitation vectors are normalized to 0.25W of total real power in the eigenchannel and zero phase for the first mode in the excitation vector. The eigenchannel coupling values from network theory are shown in Table 1 below.Separation 2.5A 5A 10A 15A 20A 25AChannel 1 0.996 0.999 0.907 0.575 0.346 0.192 Channel 2 0.996 0.998 0.839 0.554 0.316 0.162 Channel 3 0.995 0.888 0.028 0.014 0.001 0.000 Channel 4 0.994 0.751 0.016 0.004 0.000 0.000 Channel 5 0.913 0.043 0.004 0.000 0.000 0.000 Channel 6 0.731 0.019 0.000 0.000 0.000 0.000 Channel 7 0.069 0.002 0.000 0.000 0.000 0.000 Channel 8 0.011 0.000 0.000 0.000 0.000 0.000 Channel 9 0.002 0.000 0.000 0.000 0.000 0.000Table 1Attorney Docket No. 2115-008457-WO-POA
[0082] Six significant eigenchannels are seen for a 2.5A aperture separation, and maintain at least one eigenchannel with over 90% coupling out to 10A aperture separation. As distance increases, one can see only two coupled eigenchannels, as expected for the two polarizations. The null-space of Tmpredictably grows with increased separation distance, and the eigenchannels with an eigenvalue of zero are radiating eigenchannels that entirely avoid the receiving aperture.
[0083] Next, let’s consider the input impedance matrix (zin>iso) of an isolatedwaveguide aperture of equivalent dimensions, as shown in Figure 8, again using the FEKO solver.
[0084] Matching to this isolated input impedance and examining the coupling coefficients (eigenvalues) provides insight into the effects of aperture interaction. At small separation distances, one expects gains from simultaneous matching, as the aperture interaction is high. However, at larger distances, one can expect similar performance.
[0085] To verify this, compute a modified transducer gain matrix TZsousing the same Z from FEKO, but instead terminating in the Hermitian of the isolated aperture input impedance.— — —H~ ~ Zjnjso(56)
[0086] Figure 9 shows the eigenchannel coupling with simultaneous matching (42) compared with the eigenchannel coupling when matched to the isolated input impedance (56) at each separation distance. One can see that at close range, the degrees of freedom in the modal current weights allow the system to compensate for mismatch in the isolated input impedance case to still achieve near-unity coupling on the best eigenchannels. However, simultaneous matching increases the performance of the off-unity eigenchannels significantly. As distance increases such that near-unity coupling is no longer attainable, one can see gains on all usable eigenchannels with simultaneous matching. As distance increases further, the results begin to converge. This is expected, given that the aperture interaction becomes weaker, and the isolated input impedance becomes a good approximation of the true input impedance.
[0087] From this simulation, one can see that not only is the best eigenchannel improved, but the other eigenchannels improve as well. Further, Monte Carlo simulations lead us to conjecture that simultaneous Hermitian matching maximizes all p-norms of the square roots of the coupling values, given byAttorney Docket No. 2115-008457-WO-POAz I i - |P\VP(Sk=i ) with p E [l,oo]. (57)
[0088] Next, compute the field profiles at both port planes that correspond to each eigenchannel, in order to verify the network theory results through full-wave simulation. Given the port 1 current vector of the ktheigenchannel, compute the port 1 voltage vector, port 2 voltage vector, and port 2 current vector using the terminations and Z.-Jk) ->(k) -Kfc)171L1U2L2
[0089] The tangential field profiles for each mode can be computed using the analytical solution of a cylindrical waveguide mode. For the mthmode considered, with m G (1 , ..., 9), normalizing to 1 W of power flow yields,Po = 4 <W X ■ zdS (58)(59)(60)
[0090] Additionally, the modal voltage and current must be normalized to ensure the correct modal impedance, following waveguide circuit theory. Compute the following constants such that the circuit model characteristic impedance of each mode ZOnagrees with the modal wave impedance r|m.I 2Vo,m AJ-G 0) | dS (61 )■vo,m-0, m ~rlm (62)
[0091] Then compute the tangential fields of the ktheigenchannel, at both port 1 and port 2 planes in Fig. 3, using the eigenchannel voltage and current vectors above.w ^T ’pHort (A = Sm=lv0,m et,m(p' 0) (63)iw(A = Sm=l 0) (64)
[0092] Once the tangential field profiles at each port plane are computed, excite the eigenchannels by using the fields at port 1 as a source and the impedance profile at port 2 as a load. This simulation is performed using the commercial finite element electromagnetic solver COMSOL with azimuthal symmetry enforced, to allow for arbitrary field profile excitation while maintaining azimuthal invariance.Attorney Docket No. 2115-008457-WO-POA
[0093] In the COMSOL simulations used for full-wave verification, the waveguide feeds are extended 5A beyond the port planes, and magnetic current sheets are used at these offset planes within the waveguides on both the transmitting and receiving sides. Both waveguides are terminated with ports corresponding to the 9 propagating modes in the waveguide. The magnetic current sheets are then used to establish the desired field profiles (63), (64) at the original aperture port planes. This excites the eigenchannels under simultaneous Hermitian matching.
[0094] The coupling coefficient is then computed by taking the ratio of the z-directed power flow through the aperture 2 plane to the z-directed power flow through the aperture 1 plane. The coupling coefficients computed from full-wave simulation are plotted in Figure 10 along with the network theory eigenvalues (coupling coefficients from network theory). Results are shown with simultaneous Hermitian matching and matching to the isolated input impedance of the aperture.
[0095] Across the board, one sees good agreement between the full-wave and network theory results, suggesting the eigenchannels are excited. On most eigenchannels, the variation is no more than 0.02 in power coupling. The largest variation, around 0.04 to 0.05, is seen in the null eigenchannels with larger separation, namely eigenchannels 6 and 7 at 20A separation and eigenchannel 7 at 25A separation. This is explained by the difference between the MoM method used to obtain Z with FEKO and the FEM method used to excite the computed eigenchannels with COMSOL. The null eigenchannels typically form a radiating beam that avoids the second aperture. At closer separation, this beam is normally incident on the absorbing boundary of the simulation domain. As aperture separation increases, this beam is obliquely incident on the simulation boundary and carries significant power. Specular reflections from the boundary bounce directly toward aperture 2 and can result in spurious power transfer to aperture 2. As this is not a problem with MoM, the network theory eigenvalues are likely closer to the true coupling coefficients in these longer range cases with null eigenchannels. The remainder of the disclosure will focus on the case of simultaneous Hermitian matching.
[0096] Since there is no discontinuity in the design that enables polarization coupling in the multimodal 2-port, the interaction impedance matrix components (Zu , Z12, Z21 , Z22) each have a block-diagonal form. Thus, the eigenchannels each consistAttorney Docket No. 2115-008457-WO-POAof either TM or TE modes, but not both. Denote these TM channels and TE channels, respectively. Figure 11 shows the field profiles at the aperture plane of the top performing TM channel.
[0097] While the field profiles at the port consist of the 9 accessible modes at the port plane, the aperture field profiles include the evanescent modes and must be obtained from full-wave simulation after applying our method on the accessible modes. Examining the profiles, we can see that at close separations, where the channel coupling is very high, there is symmetry in the aperture 1 and aperture 2 profiles, as expected, and the phase is flipped. Figure 12 shows the field profiles at the aperture plane of the top performing TE channel.
[0098] This approach to solve for maximally coupled channels between the two apertures began with characterizing the block system impedance matrix (Z11 , Z12, Z21, Z22) between two multimodal antennas. In principle, the simultaneous matching and maximization of the transducer gain can be applied to any linear multimodal device which can be characterized by such a system impedance matrix. This is the concept known in the field as network theory, or in our case, multimodal network theory. Linear time invariant systems can be characterized by their response at the ports, which entirely determines performance. Once the system impedance matrix is obtained, in some sense, the actual device being modeled is irrelevant. Matching and the eigenproblem can be solved in the same way as we did for the coupled antenna system impedance. In electromagnetic systems, one may wish to maximize power through a segment of multimode optical fiber, where the weights on the E and H fields in each fiber are the voltages and currents at the input / output of the segment. This can also be done with a microwave transmission line network on PCB, where the modal voltages and currents are now the actual voltages and currents at the transmission line terminals. So long as there is some amount of coupling or modal mixing within the system impedance, the multimodal matching approach above can achieve higher power transfer than individually matching each modal terminal.
[0099] The techniques described herein may be implemented by one or more computer programs executed by one or more processors. The computer programs include processor-executable instructions that are stored on a non-transitory tangible computer readable medium. The computer programs may also include stored data. Non-limiting examples of the non-transitory tangible computer readable medium are nonvolatile memory, magnetic storage, and optical storage.Attorney Docket No. 2115-008457-WO-POA
[0100] Some portions of the above description present the techniques described herein in terms of algorithms and symbolic representations of operations on information. These algorithmic descriptions and representations are the means used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. These operations, while described functionally or logically, are understood to be implemented by computer programs. Furthermore, it has also proven convenient at times to refer to these arrangements of operations as modules or by functional names, without loss of generality.
[0101] Unless specifically stated otherwise as apparent from the above discussion, it is appreciated that throughout the description, discussions utilizing terms such as "processing" or "computing" or "calculating" or "determining" or "displaying" or the like, refer to the action and processes of a computer system, or similar electronic computing device, that manipulates and transforms data represented as physical (electronic) quantities within the computer system memories or registers or other such information storage, transmission or display devices.
[0102] Certain aspects of the described techniques include process steps and instructions described herein in the form of an algorithm. It should be noted that the described process steps and instructions could be embodied in software, firmware or hardware, and when embodied in software, could be downloaded to reside on and be operated from different platforms used by real time network operating systems.
[0103] The present disclosure also relates to an apparatus for performing the operations herein. This apparatus may be specially constructed for the required purposes, or it may comprise a computer selectively activated or reconfigured by a computer program stored on a computer readable medium that can be accessed by the computer. Such a computer program may be stored in a tangible computer readable storage medium, such as, but is not limited to, any type of disk including floppy disks, optical disks, CD-ROMs, magnetic-optical disks, read-only memories (ROMs), random access memories (RAMs), EPROMs, EEPROMs, magnetic or optical cards, application specific integrated circuits (ASICs), or any type of media suitable for storing electronic instructions, and each coupled to a computer system bus. Furthermore, the computers referred to in the specification may include a single processor or may be architectures employing multiple processor designs for increased computing capability.
[0104] The algorithms and operations presented herein are not inherently related to any particular computer or other apparatus. Various systems may also beAttorney Docket No. 2115-OQ8457-WO-POAused with programs in accordance with the teachings herein, or it may prove convenient to construct more specialized apparatuses to perform the required method steps. The required structure for a variety of these systems will be apparent to those of skill in the art, along with equivalent variations. In addition, the present disclosure is not described with reference to any particular programming language. It is appreciated that a variety of programming languages may be used to implement the teachings of the present disclosure as described herein.
[0105] The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the disclosure, and all such modifications are intended to be included within the scope of the disclosure.
Claims
Attorney Docket No. 2115-008457-WO-POACLAIMSWhat is claimed is:
1. A computer-implemented method for determining a maximum electromagnetic coupling of power between a source antenna and a load antenna, comprising:receiving, by a computer processor, a geometry for the source antenna, where the source antenna is represented by N modes;receiving, by the computer processor, a geometry of the load antenna, where the load antenna is represented by M modes;from the geometry for the source antenna and the geometry for the load antenna in a common electromagnetic environment, computing a system impedance matrix representing the source antenna and the load antenna and electromagnetic coupling between them using an electromagnetic field solver;defining an input impedance matrix in terms of elements of the system impedance matrix and a load impedance matrix;defining an output impedance matrix in terms of elements of the system impedance matrix and a source impedance matrix;impedance matching, by the computer processor, the source impedance matrix to the input impedance matrix and impedance matching the load impedance matrix to the output impedance matrix;defining a transducer gain for power transfer between the source antenna and the load antenna, where the transducer gain is expressed in terms of the system impedance matrix, source impedance matrix, the load impedance matrix and an excitation current for the source antenna, where the excitation current is expressed as a vector having an element for each of the N modes of the source antenna; and determining, by the computer processor, the excitation current for the source antenna by optimizing the transducer gain.
2. The method of claim 1 wherein modes of the source antenna and the load antenna are one of physical terminals connected to elements of an antenna or basis functions of the antenna.Attorney Docket No. 2115-008457-WO-POA3. The method of claim 1 further comprises computing the system impedance matrix by simulating the source antenna and the load antenna together in the common electromagnetic environment using electromagnetic field solver.
4. The method of claim 1 wherein impedance matching includes applying the multimode maximum power transfer theorem simultaneously at the ports of both the source antenna and the load antenna to find impedance matrices that maximize power.
5. The method of claim 4 further comprises setting the source impedance matrix equal to Hermitian of the input impedance matrix and setting load impedance matrix equal to Hermitian of the output impedance matrix.
6. The method of claim 1 further comprises impedance matching the source impedance matrix to the load impedance matrix using a fixed-point iteration method.
7. The method of claim 1 wherein determining the excitation current for the source antenna further comprisesexpressing the transducer gain as a generalized Rayleigh quotient, where the generalized Rayleigh quotient includes a matrix representing the source resistance and a matrix representing the load resistance;applying a transformation to the generalized Rayleigh quotient to derive a standard Rayleigh quotient;computing eigenvalues and corresponding eigenvectors for the standard Rayleigh quotient; andtransforming each eigenvector to a vector representing the excitation current that achieves the corresponding eigenvalue by inverting the transformation, where the vector represents the excitation current.
8. The method of claim 7 wherein applying a transformation to the generalized Rayleigh quotient includes factorizing the matrix representing input resistance and defining the transformation.Attorney Docket No. 2115-008457-WO-POA9. A computer-implemented method for determining a maximum electromagnetic coupling of power between a source device and a load device, comprising:computing a system impedance matrix representing the source device and the load device and electromagnetic coupling between them using an electromagnetic field solver, where the source device exhibits N modes and the load device exhibits M modes;defining an input impedance matrix in terms of elements of the system impedance matrix and a load impedance matrix;defining an output impedance matrix in terms of elements of the system impedance matrix and a source impedance matrix;simultaneously impedance matching, by a computer processor, the source impedance matrix to the input impedance matrix and impedance matching the load impedance matrix to the output impedance matrix;defining a transducer gain for power transfer between the source device and the load device, where the transducer gain is expressed in terms of the system impedance matrix, source impedance matrix, the load impedance matrix and an input for the source device, where the input is expressed as a vector having an element for each of the N modes of the source device; anddetermining, by the computer processor, the input for the source device by optimizing the transducer gain.
10. A non-transitory computer-readable medium having computerexecutable instructions that, upon execution of the instructions by a processor of a computer, cause the computer to:receive a geometry for the source antenna, where the source antenna is represented by N modes;receive a geometry of the load antenna, where the load antenna is represented by M modes;from the geometry for the source antenna and the geometry for the load antenna in a common electromagnetic environment, compute a system impedance matrix representing the source antenna and the load antenna and electromagnetic coupling between them;Attorney Docket No. 2115-008457-WO-POAimpedance match a source impedance matrix to an input impedance matrix and impedance match a load impedance matrix to an output impedance matrix, where the input impedance matrix is defined in terms of elements of the system impedance matrix and a load impedance matrix, and the output impedance matrix is defined in terms of elements of the system impedance matrix and a source impedance matrix;define a transducer gain for power transfer between the source antenna and the load antenna, where the transducer gain is expressed in terms of the system impedance matrix, source impedance matrix, the load impedance matrix and an excitation current for the source antenna, where the excitation current is expressed as a vector having an element for each of the N modes of the source antenna; and determine the excitation current for the source antenna by optimizing the transducer gain.
11. The non-transitory computer-readable medium of claim 10 wherein modes of the source antenna and the load antenna are one of physical terminals connected to elements of an antenna or basis functions of the antenna.
12. The non-transitory computer-readable medium of claim 10 wherein the computer program instructions computes the system impedance matrix by simulating the source antenna and the load antenna together in the common electromagnetic environment using electromagnetic field solver.
13. The non-transitory computer-readable medium of claim 10 wherein impedance matching includes applying the multimode maximum power transfer theorem simultaneously at the ports of both the source antenna and the load antenna to find impedance matrices that maximize power.
14. The non-transitory computer-readable medium of claim 13 wherein the computer program instructions further perform to set the source impedance matrix equal to Hermitian of the input impedance matrix and set the load impedance matrix equal to Hermitian of the output impedance matrix.
15. The non-transitory computer-readable medium of claim 10 wherein the computer program instructions further perform f impedance matching the source impedance matrix to the load impedance matrix using a fixed-point iteration method.Attorney Docket No. 2115-OQ8457-WO-POA16. The non-transitory computer-readable medium of claim 10 wherein the computer program instructions further performexpress the transducer gain as a generalized Rayleigh quotient, where the generalized Rayleigh quotient includes a matrix representing the source resistance and a matrix representing the load resistance;apply a transformation to the generalized Rayleigh quotient to derive a standard Rayleigh quotient;compute eigenvalues and corresponding eigenvectors for the standard Rayleigh quotient; andtransform each eigenvector to a vector representing the excitation current that achieves the corresponding eigenvalue by inverting the transformation, where the vector represents the excitation current.