Abstract

Modes generally provide an economical description of waves, reducing complicated wave functions to finite numbers of mode amplitudes, as in propagating fiber modes and ideal laser beams. But finding a corresponding mode description for counting the best orthogonal channels for communicating between surfaces or volumes, or for optimally describing the inputs and outputs of a complicated optical system or wave scatterer, requires a different approach. The singular-value decomposition approach we describe here gives the necessary optimal source and receiver “communication modes” pairs and device or scatterer input and output “mode-converter basis function” pairs. These define the best communication or input/output channels, allowing precise counting and straightforward calculations. Here we introduce all the mathematics and physics of this approach, which works for acoustic, radio-frequency, and optical waves, including full vector electromagnetic behavior, and is valid from nanophotonic scales to large systems. We show several general behaviors of communications modes, including various heuristic results. We also establish a new “M-gauge” for electromagnetism that clarifies the number of vector wave channels and allows a simple and general quantization. This approach also gives a new modal “M-coefficient” version of Einstein’s A&B coefficient argument and revised versions of Kirchhoff’s radiation laws. The article is written in a tutorial style to introduce the approach and its consequences.

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  119. W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
    [Crossref]
  120. Specifically, I presume basic algebra and calculus, elementary real analysis (including convergence and limits), basic linear algebra including matrices with eigenvectors and eigenvalues, differential equations including elementary partial differential equations and eigenfunctions and eigenvalues, integral equations at least up to elementary Green’s functions, vectors, elementary vector calculus, basic notions of sets including the usual sets of numbers (integer, real, complex), basic wave equations, and electromagnetism up to and including Maxwell’s equations in differential form. Such mathematics is covered well by a text such as [127]. I specifically do not presume any knowledge of functional analysis (which would be relatively uncommon for physical scientists and engineers), and I do not require advanced knowledge of electromagnetism, such as the use of vector potentials and gauges. The advanced electromagnetism and the functional analysis needed are covered later in this work and in Ref. [122].
  121. The various mathematical properties of finite matrix eigenproblems can, however, be deduced from the results of functional analysis; a finite matrix is then a special case.
  122. D. A. B. Miller, “An introduction to functional analysis for science and engineering,” arXiv:1904.02539 (2019).
  123. The pulses in a general time-dependent field would have to be square-integrable, but physically that essentially corresponds to finite energy, which we would expect anyway.
  124. B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
    [Crossref]
  125. D. V. Reddy and M. G. Raymer, “High-selectivity quantum pulse gating of photonic temporal modes using all-optical Ramsey interferometry,” Optica 5, 423–428 (2018).
    [Crossref]
  126. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994).
  127. G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).
  128. It is convenient algebraically to pretend that the time dependence of the wave is of the form exp(−iωt) and to work with complex amplitudes. Classical waves are, however, real, but we can always get back to that by adding the complex conjugate at the end.
  129. We will introduce a somewhat more general definition of adjoint operators below, but for matrices, this definition is sufficient.
  130. G. Strang, Linear Algebra and its Applications, 3rd ed. (Harcourt Brace Jovanovich, 1988).
  131. R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. (Cambridge University, 2013).
  132. We postpone the definition of a compact operator since this is, unfortunately, rather technical. However, as we will argue below, the coupling operators associated with wave equations are Hilbert–Schmidt operators, which are all compact. The important results we discuss here are therefore going to apply very generally in wave problems.
  133. To be quite technically correct here, if any of the eigenvalues is zero, then the set is not necessarily complete, but it can always be extended, for example, by a process such as Gram–Schmidt orthogonalization, to be complete.
  134. If NS and NR are not equal, then all of the “extra” singular values formally associated with the larger of the two matrices are identically zero. That is, if Nlarge is the larger of NS and NR and Nsmall is the smaller of NS and NR, then all of the “extra” singular values sj from j=Nsmall+1 to j=Nlarge are zero. Formally, also, we are free to generate orthogonal eigenfunctions associated with these extra “0” singular values that are orthogonal to all of the first Nsmall eigenfunctions and to each other, by some process such as Gram–Schmidt orthogonalization. These “extra” eigenfunctions do not participate at all in communicating with waves between the sources and the receivers and, other than being orthogonal to one another and to the first Nsmall eigenfunctions, their choice is relatively arbitrary because they are also solutions of a degenerate eigenproblem, all sharing the same eigenvalue of zero.
  135. At least, the sets are complete for discussing communications between the volumes, and they can be extended beyond that if necessary. See notes [133] and [134].
  136. Note, incidentally, that such degeneracies are always finite for eigensolutions of compact Hermitian operators.
  137. Note that, quite generally, S=Tr(GSR†GSR)=Tr(GSRGSR†). The trace of a matrix does not depend on the complete orthonormal basis set(s) used to represent it and is therefore also the sum of the eigenvalues of the matrix [because we could represent the matrix on the eigen (or SVD) basis set(s)].
  138. Note that the eigenvectors of both the GSR†GSR and GSRGSR† operators are each arbitrary within a unit phase factor (e.g., of the form exp(iθ) for some angle θ). In practice, this may mean that we make some choice of phase for each of the eigenvectors that is convenient for us, and if or when we need the singular values sj rather than just the squared modulus |sj|2, we can formally establish the phase factors by computing, say, GSR|ψSj⟩≡sj|ϕRj⟩, as in Eq. (31).
  139. Somewhat higher numerical precision is needed to see this orthogonality precisely.
  140. Other configurations of the phase shifters in the block are possible and ultimately equivalent; for example, we could use one phase shifter on the top arm of the interferometer to control the split ratio and a second phase shifter on, say, the top output waveguide to control an additional phase. See [24–28].
  141. For example, as discussed in [24] and [25], for an input signal in the top ESIn1 guide, we can progressively set the S11 and S22 blocks and the S31 phase shifter to give the desired power splitting and output phases for the first channel (i.e., the vector |ψS1⟩ of output amplitudes). For the second channel, with input in the middle ESIn2 guide, because we know the settings of the S11, S22, and S31 elements, we can readily calculate what outputs are required from the S12 and S22 elements to achieve the |ψS2⟩ set of output amplitudes at the right and hence we can calculate how to set those elements. We might think that we do not have enough elements to allow us to specify |ψS2⟩, but we do because it is guaranteed to be orthogonal to |ψS1⟩, which reduces the number of required independent parameters by two. For the final ESIn3 input, because |ψS3⟩ must be orthogonal to both |ψS2⟩ and |ψS1⟩, the only remaining independent parameter to set is the phase shift.
  142. The half-wavelength spacing clarifies the behavior of the resulting waves because at spacings of half a wavelength or shorter, additional “diffraction orders” are eliminated, so there are no spurious additional beams to confuse the pictures of the waves.
  143. The phase of the source and receiver modes is arbitrary, as is generally the case with such eigenmodes; multiplying an eigenmode by a complex factor still leaves it as a solution of the same (linear) eigenproblem. Furthermore, these eigenproblems only give us |sj|2, which similarly leaves us free to choose the phase of sj to be whatever is convenient.
  144. Because these sources are actually in three-dimensional space, they are also transmitting in the directions in and out of the plane, and, indeed, actually equally well in all directions in the horizontal plane.
  145. D. A. B. Miller, “Huygens’s wave propagation principle corrected,” Opt. Lett. 16, 1370–1372 (1991).
    [Crossref]
  146. It is, incidentally, interesting to see how and why these pairs of sources work. Note, first, that in the calculations to generate the modes we made no prescription about the relative amplitudes and phases of the two different lines of sources. The resulting amplitudes and phases result entirely from the solution of the eigenvalues and eigenfunctions of the relevant matrix (GSRGSR† or GSR†GSR). In establishing the best possible source amplitudes, the numerical solution has “found” an approach that can be called a “spatiotemporal dipole” [145]. An ideal such spatiotemporal dipole would have equal and opposite amplitude for the two sources in the dipole (one on the “left” and one on the “right”) but with a phase lag on the “left” source that corresponds to the time taken for the wave to travel between the two sources in the pair. That leads to at least partially constructive addition on the “right” but destructive interference on the left. In this case, we see numerically that the amplitudes of the left and right sources in each pair are indeed approximately equal in magnitude, and the left source does indeed lag to the right by approximately the right phase [90° (π/2) for sources separated by a quarter wave]. Note again that the solution of this problem “found” this desirable behavior automatically; we did not “tell” the mathematics to find such spatiotemporal dipole solutions. Such spatiotemporal dipoles are also a particularly elegant way to restate Huygens’ principle [145], giving much better numerical results than the simple point sources of Huygens’ original proposal and eliminating unphysical backward waves.
  147. So far, for simplicity, we presented SVD with equal numbers of source and receiver points, which resulted in a square matrix for GSR. In fact, though, such equal numbers are not necessary for SVD, and, correspondingly, SVD can be performed on a matrix that is not square. In our present case, though we have doubled the number of source points to NS=18, we can keep the number of receiving points at NR=9. In such a case the matrix GSR is a 9×18 matrix rather than a square one. In this case, the matrix GSR†GSR is an 18×18 matrix, whereas the matrix GSRGSR† is 9×9, which might seem to give a contradiction. Solving the GSR†GSR eigenproblem would give 18 eigenfunctions, whereas solving the GSRGSR† eigenproblem would give only nine. The resolution of this paradox is that the eigenvalues (and the singular values) for the additional nine eigenfunctions in the GSR†GSR case are mathematically identically zero [134]. The corresponding source functions have mathematically absolutely no coupling strength to the receivers. In our numerical calculations, the power coupling strengths of these additional modes are approximately 10−17 times as small rather than being exactly zero, with this finite but small value presumably reflecting rounding errors and limitations in the numerical calculations.
  148. The mode in Fig. 7(a) is actually also the second most strongly coupled mode, though its coupling is smaller than the most strongly coupled mode only by a very small amount. The other two strongly coupled modes are analogous to those of Fig. 6(a) (a “two-bumped” mode) and Fig. 6(c) (a “three-bumped” mode), and these have very similar coupling strengths to one another in this case also. The percentages of the corresponding sum rule S for each of these three modes for the source and receiver arrangement of Fig. 7 are ∼28.04%, ∼28.51%, and ∼26.24%, for the “one-,” “two-,” and “three-” “bumped” modes, respectively.
  149. Note, incidentally, that these power coupling strengths |sj|2 are not formal power coupling efficiencies between sources and receivers, nor are they necessarily even proportional to the power coupling efficiencies. We are not formally evaluating the total power emitted by the sources. These |sj|2 are the relative powers in each beam when starting with source functions of unit amplitude, but those unit amplitudes do not necessarily all correspond to unit emitted power.
  150. D. Gabor, “Light and information,” Prog. Opt. 1, 109–153 (1961).
    [Crossref]
  151. There is a small imaginary component left near the ends of the line of receiver points, so the wave is not exactly confocally curved there, though the real part is still quite a good representation of the overall wave amplitude there.
  152. We are introducing the term “paraxial degeneracy” here.
  153. Note that the sum rule S is different for each of these cases, and it would be wrong to conclude that the coupling strengths are generally reducing in magnitude as we make the volume thicker, even with non-uniform shapes. Generally, increasing the thickness (while correspondingly increasing the number of points in the volume) increases the absolute coupling strength. As we increase thickness non-uniformly, as in ellipsoidal source volumes, for some of the modes, the increase in coupling strength is more than for others.
  154. If we increase the length of the line of receivers and correspondingly increase the separation between the sources and the receivers, so the angle subtended by the source line at the receivers is essentially constant, then the “knee” in the curves here moves closer to NHy—that is, the factor that here is 0.985 moves closer to 1. The form of the curve, explicitly including the exponential decay rate, does not change, however, with the singular values falling off exponentially with the same exponent.
  155. In this rationalization, we presume that we can approximately “factorize” the modes into a product of “horizontal” and “vertical” mode forms, like those seen with “line” sources and receivers. Up to n≃NH, both the horizontal and vertical forms are for modes below the corresponding NHx and NHy limits. However, for n>NH, one or the other of the horizontal or vertical forms must exceed its corresponding NHx and NHy limits. So, there will be a set of NHx “horizontal” modes that correspond to the first “vertical” mode past the limit, and similarly a set of NHy modes that correspond to the first “horizontal” mode past the limit. So we expect to see a “step” with ≈NHx+NHy modes with approximately equal singular values. A similar argument for successive weaker modes in one or the other direction leads to a subsequent step, and so on. Because there is a number of such modes on each step that is therefore proportional (in this square case) to NHx=NHy=NH, we divide by NH in the exponential. Of course, this is not quite a complete counting of all the possible weakly coupled modes, because there will also be modes in which both the “horizontal” and “vertical” modes are both “weak,” so this rationalization is not a complete description, but it does give some sense as to why we can see “steps” and the NH factor in the denominator in the approximate exponential.
  156. The general behavior of singular values for prolate spheroidal functions is well known [157] and expands this discussion for the weakly coupled values in a general “Fourier-transform” approach, showing that the number of “degrees of freedom” increases only logarithmically as the minimum acceptable singular value is decreased {[157], Eq. (2)}, which is consistent with an exponentially decaying strength of the singular values. Reference [158] extends this to more dimensions. Insofar as these Fourier-transform approaches are valid, which may hold in the limit of large structures separated by even larger distances, they give some explanation for this phenomenon.
  157. H. J. Landau and H. Widom, “Eigenvalue distribution of time and frequency limiting,” J. Math. Anal. Appl. 77, 469–481 (1980).
    [Crossref]
  158. M. Franceschetti, “On Landau’s eigenvalue theorem and information cut-sets,” IEEE Trans. Inf. Theory 61, 5042–5051 (2015).
    [Crossref]
  159. C. A. Balanis, ed., Modern Antenna Handbook (Wiley, 2008).
  160. We used a “spiral” approach to obtain an approximately uniform distribution of points on the spherical surface. See E. B. Saff and A. B. J. Kuijlaars, “Distributing many points on a sphere,” The Mathematical Intelligencer 19, 5–11 (1997), in our case using a “golden section” ratio angular increment π(3−5). A pseudo-code version of this algorithm is as follows for distributing Nsp points on a sphere of unit radius, returning arrays x, y, and z of the corresponding x, y, and z values for each point. Here, “;” is a statement separator, and xk is the kth element of the array x. g≔0; dz≔2/Nsp; s≔1−dz/2; dg≔π(3−5); For k≔1 to Nsp r≔1−s2;xk≔r cos(g); yk≔r sin(g);zk≔s; s≔s−dz; g≔g+dg.
  161. M. Bertero and E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis,” Opt. Acta 29, 727–746 (1982).
    [Crossref]
  162. The modal amplitudes used were chosen somewhat arbitrarily, but with suitable values for modes 12, 13, and 14, to illustrate the key points without also showing extreme behavior (such as very large source amplitudes at specific sources). The actual modal amplitudes in this example for the desired received field are (to three significant figures), in order for modes 1 to 14, 0.183, 0.143, 0.148, 0.346, 0.445, 0.207, 0.188, 0.395, 0.469, 0.198, 0.247, 0.104, 0.178, and 0.099. The phases of the modes were chosen randomly with a uniform distribution over all phases from −π to π; explicitly, those phases, for modes 1 to 14, are π times the following values: 0.823, 0.912, −0.668, −0.235, −0.141, −0.353, 0.678, 0.749, −0.919, 0.711, −0.098, −0.170, 0.935, and 0.325. (All numbers are quoted to three significant figures, though the actual values used had higher precision.)
  163. J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14, 083001 (2012).
    [Crossref]
  164. G. de Villiers and E. R. Pike, The Limits of Resolution (Taylor and Francis, 2016).
  165. K. Piché, J. Leach, A. S. Johnson, J. Z. Salvail, M. I. Kolobov, and R. W. Boyd, “Experimental realization of optical eigenmode super-resolution,” Opt. Express 20, 26424–26433 (2012).
    [Crossref]
  166. M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
    [Crossref]
  167. F. M. Dickey, L. A. Romero, J. M. DeLaurentis, and A. W. Doerry, “Super-resolution, degrees of freedom and synthetic aperture radar,” IEE Proc. 150, 419–429 (2003).
    [Crossref]
  168. G. Lerosey, F. Lemoult, and M. Fink, “Beating the diffraction limit with positive refraction: the resonant metalens approach,” in Plasmonics and Super-Resolution Imaging, Z. Liu, ed. (Pan Stanford, 2017), pp. 33–90.
  169. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, 1978).
  170. D. Porter and D. S. G. Stirling, Integral Equations: A Practical Treatment, from Spectral Theory to Applications (Cambridge University, 1990).
  171. J. K. Hunter and B. Nachtergaele, Applied Analysis (World Scientific, 2001).
  172. In (IP2), for good reason, we choose a notation convention here that is the other way around from most (but not all) mathematics texts. Common mathematical notation for (IP2) would have (aγ,α)=a(γ,α), which, with (IP3), would give (γ,aα)=a*(γ,α). Our choice corresponds better with the order we encounter in our “algebraic shift” to Dirac notation, and gives a natural form of the associative property of multiplication as in matrix-vector notation.
  173. This term “underlying inner product” is one that I am introducing here for clarity.
  174. Note that this is technically a reuse of a notation; we already used (α,β) with such ordinary braces for the inner product. Such reuse is unfortunately rather common in mathematical texts.
  175. Unfortunately, this “infinitely long” aspect of a given sequence may well not be stated clearly or explicitly in functional analysis.
  176. In a metric space with a metric d(α,β), a sequence (αn) is said to be Cauchy (or to be a Cauchy sequence) if for every real number ε>0 (no matter how small) there is a number N (a positive integer or natural number) such that, for every m,n>N, d(αm,αn)<ε.
  177. This use of “complete” in a “complete set” is different from the idea of a “complete” space; this confusion is unfortunate, but is unavoidable because of common usage.
  178. Technically, the supremum is the smallest number that is greater than or equal to all the numbers being considered.
  179. In a notation like this with this “dot”, it is best to view these inner-product operations as “waiting to happen”; just how much of the inner-product operation we are effectively writing down here can be somewhat vague in mathematics texts. However, we will take the approach that both any “operator weighting” and any integral for the inner product are “waiting to be applied” and in that sense are not yet part of this expression.
  180. In its more common use in quantum mechanics, Dirac notation is not required to deal with the sophistication of different underlying inner products, though we see here that, with careful definitions, it can handle this extension.
  181. Quite generally, a form such as |βj⟩2⟨αk|1 is an outer product. In contrast to the inner product, which produces a complex number, and which necessarily only involves vectors in the same Hilbert space, the outer product generates a matrix from the multiplication in “column-vector row-vector” order, and can involve vectors in different Hilbert spaces.
  182. The same problem does not arise in finite-dimensional spaces; if we construct an infinitely long sequence made up from just the finite number of basis vectors in the space, we will have to repeat at least one of the basis vectors an infinite number of times, which gives us at least one convergent subsequence—the (sub)sequence consisting of just that basis vector repeated an infinite number of times. In fact, we can prove [122] that it is sufficient that an operator has finite-dimensional range for it to be compact. A corollary is that operators described by finite matrices are compact.
  183. Note that there is some variation in notation in mathematics texts. Kreyszig [169] uses this definition for a positive operator, for example, and if the “≥” sign is replaced by a “>” sign in Eq. (133), he would then call the operator positive-definite. Others, however, such as [170], would give Eq. (133) as defining a non-negative operator, using “positive operator” only if the “≥” sign is replaced by a “>” sign.
  184. Both “operator-weighted inner product” and “transformed inner product” are terms we are adding here; I know of no other standard names for these concepts.
  185. Incidentally, note that, unlike many basis transformation, there is no requirement here that this transform is unitary.
  186. In fact, it is not even necessary with such a “1/r2” integrand that the volumes do not overlap; the result of such an integral will be finite even if the resulting “1/r2” singularity is included. See the discussion in [187], p. 140 and p. 173.
  187. G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics (Springer, 2002).
  188. Here, as noted in the discussion after Eq. (105), we have made the notational choice to leave the integrals over rR and rS out of this part of the mathematics, including them later when we perform the actual inner products.
  189. C. Huygens, Traité de la Lumiere (Leyden, 1690). [English translation by S. P. Thompson, Treatise on Light (Macmillan, 1912).]
  190. The full solution for scalar waves requires two kinds of sources, which can be written as point sources and spatial dipoles, but can also be written as spatio-temporal dipoles [145]; only one kind of such spatio-temporal dipole is typically required, however, for “slowly varying” wavefronts, allowing a return to a simple view of effective sources on the wavefront, and hence a simple Green’s function with an obliquity factor included.
  191. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  192. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [Crossref]
  193. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
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    [Crossref]
  195. A. Karoui and I. Mehrzi, “Asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions,” Appl. Math. Comput. 218, 10871–10888 (2012).
    [Crossref]
  196. A. Karoui and I. Mehrzi, “Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions,” J. Comp. Appl. Math. 233, 315–333 (2009).
    [Crossref]
  197. G. Walter and T. Soleski, “A new friendly method of computing prolate spheroidal wave functions and wavelets,” Appl. Comput. Harmon. Anal. 19, 432–443 (2005).
    [Crossref]
  198. C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).
  199. Using a notation after [198], the nth such eigenfunction solution in the y direction in a given source or receiver aperture is of the form S0n(cL,ky/L), where the parameter cL=(π/2)NPy, with NPy being the paraxial heuristic number in the y direction. A similar set of solutions will exist in the x direction.
  200. If we try to derive solutions with transverse boundaries at infinity using integral equations (as in our Green’s function SVD approaches), the corresponding singular values are all identical (there is no “loss” in the propagation because no wave “misses” the receiver space), so the eigensolutions are completely degenerate and any orthogonal basis is equally good, so the solution becomes mathematically trivial. Hermite–Gaussians will be solutions in one direction, for example, but so will any other set of complete functions. There is also a physical contradiction in such solutions without finite boundaries because we are violating the paraxial approximation by allowing the boundaries to extend arbitrarily in the transverse direction.
  201. Beams with a specific “orbital” angular momentum correspond to a phase variation in azimuthal angle ϕ with an integer “quantum number” m in the form exp(imϕ). If the solutions with exp(imϕ) and exp(−imϕ) are degenerate, then we are free to construct the linear combinations with phase variations of the form cos(mϕ)=(1/2)[exp(imϕ)+exp(−imϕ)] and sin(mϕ)=(1/2i)[exp(imϕ)−exp(−imϕ)]. These new solutions, each being equal sums of solutions with equal but opposite “orbital” angular momentum, have zero “orbital” angular momentum. It is then a matter of taste whether we want to work with positive and negative m and exp(imϕ) solutions (with net “orbital” angular momentum), or positive m with cos(mϕ) and sin(mϕ) solutions (with no net “orbital” angular momentum). The total number of orthogonal functions available up to some specific |m| is exactly the same.
  202. M. Tamagnone, C. Craeye, and J. Perruisseau-Carrier, “Comment on ‘Encoding many channels on the same frequency through radio vorticity: first experimental test’,” New J. Phys. 14, 118001 (2012).
    [Crossref]
  203. M. Tamagnone, C. Craeye, and J. Perruisseau-Carrier, “Comment on ‘Reply to Comment on’ Encoding many channels on the same frequency through radio vorticity: first experimental test’,” New J. Phys. 15, 078001 (2013).
    [Crossref]
  204. R. Gaffoglio, A. Cagliero, G. Vecchi, and F. P. Andriulli, “Vortex waves and channel capacity: hopes and reality,” IEEE Access 6, 19814–19822 (2017).
    [Crossref]
  205. J. Xu, “Degrees of freedom of OAM-based line-of-sight radio systems,” IEEE Trans. Antennas Propag. 65, 1996–2008 (2017).
    [Crossref]
  206. C. Shi and X. Zhang, “Reply to Miller: misunderstanding and mix-up of acoustic and optical communications,” Proc. Natl. Acad. Sci. USA 114, E9757–E9758 (2017).
    [Crossref]
  207. In my opinion, [206] is incorrect in every substantial criticism made of my response [23] to those authors’ earlier paper on acoustic “orbital” angular momentum beams [22]. I used the term “optical angular momentum,” which is one of the terms in the field (see [31]), and I have not confused acoustic and optical communication. In my opinion, my paper [23] stands correct as written. See [208] for specific comments.
  208. I have calculated with the scalar Green’s function, which is a first approximation in optics, but is the right approach for these acoustic waves, and it is acoustic channels that I calculated. With my approach, using communications modes, I achieved more channels, with fewer transmitters and receivers, and, contrary to these authors’ statements, my approach has no crosstalk in principle, not the −7.7  dB asserted by these authors (the −7.7  dB refers to channel strengths, not crosstalk).
  209. Just to construct an orthogonal basis set AωMbn(rR) in VR, it is not strictly necessary to choose subsequent Jωbn(rS) to be orthogonal to all preceding ones Jωbm(rS); linear independence would be sufficient to allow construction of orthogonal AωMbn(rR). But choosing the Jωbn(rS) to be mutually orthogonal means that in this process we also usefully generate an orthogonal basis for the source functions.
  210. Note that there are few restrictions on what form these current sources take—they are not functions that have to obey Maxwell’s equations, for example. Overall, we would require conservation of charge, but that will be automatic if these are monochromatic, and therefore purely oscillatory, functions. These orthogonal current sources could be as simple as small uniform patches on a surface. We would of course have to choose vector directions for such current patches, and if we want the resulting set to be complete, we should include versions with the currents in three vector directions that are at right angles.
  211. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).
  212. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Oxford University, 1995).
  213. H. Haken, Light (North-Holland, 1981), Vol. 1.
  214. O. L. Brill and B. Goodman, “Causality in the Coulomb gauge,” Am. J. Phys. 35, 832–837 (1967).
    [Crossref]
  215. A first problem with the Coulomb gauge is that the equation for the scalar potential is unphysical in that any change in charge density in any region of space results in instantaneous changes of the potential ΦC everywhere in space. This apparent inconsistence with relativity does not result in actual violations of the velocity of light propagation of the fields E and B [214,248], but it is at least awkward. A second problem with the Coulomb gauge is that in wave problems we typically proceed by separation into “longitudinal” and “transverse” current densities. In free space, with no actual current densities anywhere in space, this causes no additional problems, but if there is indeed any current density at any point or finite region in space (and we expect to have source densities in our problems), the resulting longitudinal and transverse effective source current densities actually extend through all space [244].
  216. Note, incidentally, that, though we are just using the “exp(−iωt)”AωMj(rR) parts in performing this calculation of energy, the resulting energy is the energy of the total real field AR(rR,t) because of the way we set up the energy inner products.
  217. We have drawn this with grating couplers in vertical lines at the ends of the waveguides, but this is just an example. We can have any optics between the “source” waveguides on the left and the “receiver” waveguides on the right, and in two-dimensional arrangements, not just these vertical lines.
  218. For this particular approach to work, the optical system has to be reciprocal.
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  237. We need to retain terms up to ∼ε2 to get a non-zero result in the algebra.
  238. (|α⟩⟨β|)†=(⟨β|)†(|α⟩)†=|β⟩⟨α|.
  239. There is a minor formal point that we have only proved these results for functions corresponding to non-zero eigenvalues (or singular values). For our communications problems, we can disregard any channels with zero coupling strength, so functions associated with zero singular values are of no interest. As far as the Hilbert spaces are concerned, we could either restrict them to using as basis sets only those eigenfunctions corresponding to non-zero singular values or extend the basis sets by a process such as Gram–Schmidt orthogonalization to construct basis sets for the larger spaces. None of those additional basis functions in either space will participate in the “communication” between the spaces, so it is of no consequence in our problem which of these formal approaches we use.
  240. A unitary matrix B is one for which B†B=Iop, the identity operator. Quite generally, for any complete set {|γp⟩}, we can write Iop=∑p|γp⟩⟨γp| (such an expansion operating on any function in the space simply returns the same function). For our matrix here, we have U†U=(∑p|ψp⟩⟨γp|)†(∑q|ψq⟩⟨γq|)=∑p,q|γp⟩⟨ψp|ψq⟩⟨γq|=∑p|γp⟩⟨γp|=Iop, and similarly for V†V.
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  246. The subscript “C” is already in use for the Coulomb gauge, and the next letter (“o”) in the word “communications” has too many other uses, so we use the third letter, and continue in using uppercase letters for gauges, leading to the subscript “M” for this gauge (which also distinguishes it from the use of “m” for “magnetic,” as in Jm earlier).
  247. The gauge in which the scalar potential is set completely to zero is known as the Hamiltonian or temporal gauge (see [149]). Here we retain a fixed scalar potential [as in Eq. (G30)] to deal with the static fields, which makes this M-gauge different from that Hamiltonian or temporal gauge.
  248. It is possible with this M-gauge to write scalar wave equations for each of the vector components of AM, and driven by the corresponding vector components of an effective current density JM. However, the physical interpretation of this effective current density is somewhat involved and not particularly illuminating.
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  253. Such behavior is, of course, also well described by tensors; a dyadic can in general just be viewed as a second rank tensor for a three-dimensional space and a dyadic can be written as a 3×3 matrix, with the three dimensions corresponding to three orthogonal unit vector directions.
  254. Dyadic notation can also be viewed as an extension of vector and vector calculus notation, allowing obvious generalization of theorems and identities in vector and vector calculus algebra.
  255. We presume that ∇δ(r−r′) is meaningful, which it will be if we approximate the delta function by an appropriate but very “sharp” function with continuous derivatives and formally take the limit as the function becomes “sharper.”
  256. There is a very subtle point about Green’s functions for such a vector wave equation if we are looking at waves at points where there are also source current densities. In that case another “depolarization dyadic” term has to be added [257], equivalent to including an additional delta function term in the Green’s function itself. Since our sources are in one volume and the waves of interest to us are in another, we do not have to consider this term here, however.
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  258. This statement with an amplitude Eoω(r) or Boω(r) is consistent with E(r,t)=Re(Eoω(r)exp[i(ωt+θe)]) or B(r,t)=Re(Boω(r)exp[i(ωt+θm)]). Such statements are common in electromagnetism textbooks (e.g., [242,243]).
  259. Note too that θe and θm may vary with r.
  260. The time average of cos2(ωt+θ) over a cycle is ½.
  261. This form is also implicit for any particular frequency if the fields are Fourier-transformed and the frequencies in the Fourier transform are presumed to run over positive and negative values [241].
  262. We could continue to write this same inner product as an operator-weighted inner product by multiplying out U¯¯=(U)†U (which gives a dyadic operator as a result, hence the notation). In that case, we could formally write the inner product at some time t as (μ,η)U¯¯≡(μ,U¯¯η)≡∫Vμ*(r,t)·U¯¯·η(r,t)d3r. This operator U¯¯ could be written as a 3×3 matrix operating to the right on the mathematical column vector of components of the vector potential field η, and on the left on the Hermitian adjoint of the mathematical vector of components of the vector potential field μ. However, that requires that we have the unusual situation of some derivatives operating to the left instead of to the right; that is mathematically straightforward, but it requires a correspondingly unusual notation, so for simplicity we omit it.
  263. Note the similarity of this expression to the classical one in Eq. (189). Indeed, if we were to “symmetrize” aj*(t)aj(t)→(1/2)[aj(t)aj*(t)+aj*(t)aj(t)], rewrite the “a’s” as operators, and postulate the commutation relation Eq. (367), we would get Eq. (369).

2019 (4)

M. A. Maisto and F. Munno, “The role of diversity on linear scattering operator: the case of strip scatterers observed under the Fresnel approximation,” Electronics 8, 113 (2019).
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R. Solimene, M. A. Maisto, and R. Pierri, “Sampling approach for singular system computation of a radiation operator,” J. Opt. Soc. Am. A 36, 353–361 (2019).
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H. Yılmaz, C. W. Hsu, A. Yamilov, and H. Cao, “Transverse localization of transmission eigenchannels,” Nat. Photonics 13, 352–358 (2019).
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D. Marpaung, J. Yao, and J. Capmany, “Integrated microwave photonics,” Nat. Photonics 13, 80–90 (2019).
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2018 (7)

P. L. Mennea, W. R. Clements, D. H. Smith, J. C. Gates, B. J. Metcalf, R. H. S. Bannerman, R. Burgwal, J. J. Renema, W. S. Kolthammer, I. A. Walmsley, and P. G. R. Smith, “Modular linear optical circuits,” Optica 5, 1087–1090 (2018).
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X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
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J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
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R. Tang, T. Tanemura, S. Ghosh, K. Suzuki, K. Tanizawa, K. Ikeda, H. Kawashima, and Y. Nakano, “Reconfigurable all-optical on-chip MIMO three-mode demultiplexing based on multi-plane light conversion,” Opt. Lett. 43, 1798–1801 (2018).
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N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophotonic processors,” Optica 5, 1623–1631 (2018).
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D. V. Reddy and M. G. Raymer, “High-selectivity quantum pulse gating of photonic temporal modes using all-optical Ramsey interferometry,” Optica 5, 423–428 (2018).
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P. J. Winzer, D. T. Neilson, and A. R. Chraplyvy, “Fiber-optic transmission and networking: the previous 20 and the next 20 years [Invited],” Opt. Express 26, 24190–24239 (2018).
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2017 (17)

J. M. Kahn and D. A. B. Miller, “Communications expands its space,” Nat. Photonics 11, 5–8 (2017).
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D. A. B. Miller, L. Zhu, and S. Fan, “Universal modal radiation laws for all thermal emitters,” Proc. Natl. Acad. Sci. USA 114, 4336–4341 (2017).
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M. L. Wang and A. Arbabian, “Exploiting spatial degrees of freedom for high data rate ultrasound communication with implantable devices,” Appl. Phys. Lett. 111, 133503 (2017).
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C. Shi, M. Dubois, Y. Wang, and X. Zhang, “High-speed acoustic communication by multiplexing orbital angular momentum,” Proc. Natl. Acad. Sci. USA 114, 7250–7253 (2017).
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D. A. B. Miller, “Better choices than optical angular momentum multiplexing for communications,” Proc. Natl. Acad. Sci. USA 114, E9755–E9756 (2017).
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A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling light—automatically undoing strong mixing between modes,” Light Sci. Appl. 6, e17110 (2017).
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D. A. B. Miller, “Setting up meshes of interferometers—reversed local light interference method,” Opt. Express 25, 29233–29248 (2017).
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R. N. S. Suryadharma, M. Fruhnert, C. Rockstuhl, and I. Fernandez-Corbaton, “Singular-value decomposition for electromagnetic-scattering analysis,” Phys. Rev. A 95, 053834 (2017).
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M. Mazilu, T. Vettenburg, M. Ploschner, E. M. Wright, and K. Dholakia, “Modal beam splitter: determination of the transversal components of an electromagnetic light field,” Sci. Rep. 7, 9139 (2017).
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R. Gaffoglio, A. Cagliero, G. Vecchi, and F. P. Andriulli, “Vortex waves and channel capacity: hopes and reality,” IEEE Access 6, 19814–19822 (2017).
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J. Xu, “Degrees of freedom of OAM-based line-of-sight radio systems,” IEEE Trans. Antennas Propag. 65, 1996–2008 (2017).
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C. Shi and X. Zhang, “Reply to Miller: misunderstanding and mix-up of acoustic and optical communications,” Proc. Natl. Acad. Sci. USA 114, E9757–E9758 (2017).
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D. W. Prather, S. Shi, G. J. Schneider, P. Yao, C. Schuetz, J. Murakowski, J. C. Deroba, F. Wang, M. R. Konkol, and D. D. Ross, “Optically upconverted, spatially coherent phased-array-antenna feed networks for beam-space MIMO in 5G cellular communications,” IEEE Trans. Antennas Propag. 65, 6432–6443 (2017).
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Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
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N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
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A. Boniface, M. Mounaix, B. Blochet, R. Piestun, and S. Gigan, “Transmission-matrix-based point-spread-function engineering through a complex medium,” Optica 4, 54–59 (2017).
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S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
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2016 (12)

J. Bosch, S. A. Goorden, and A. P. Mosk, “Frequency width of open channels in multiple scattering media,” Opt. Express 24, 26472–26478 (2016).
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M. Fink, “From Loschmidt daemons to time-reversed waves,” Phil. Trans. Roy. Soc. A 374, 20150156 (2016).
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D. Akbulut, T. Strudley, J. Bertolotti, E. P. A. M. Bakkers, A. Lagendijk, O. L. Muskens, W. L. Vos, and A. P. Mosk, “Optical transmission matrix as a probe of the photonic strength,” Phys. Rev. A 94, 043817 (2016).
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A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
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C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach-Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
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T. Tudor, “Nonnormal operators in physics, a singular-vectors approach: illustration in polarization optics,” Appl. Opt. 55, B98–B106 (2016).
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W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
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J. Carpenter, B. J. Eggleton, and J. Schröder, “Complete spatiotemporal characterization and optical transfer matrix inversion of a 420 mode fiber,” Opt. Lett. 41, 5580–5583 (2016).
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M. Lee, M. A. Neifeld, and A. Ashok, “Capacity of electromagnetic communication modes in a noise-limited optical system,” Appl. Opt. 55, 1333–1342 (2016).
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M. Chen, K. Dholakia, and M. Mazilu, “Is there an optimal basis to maximise optical information transfer?” Sci. Rep. 6, 22821 (2016).
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A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a 4 × 4-port universal linear circuit,” Optica 3, 1348–1357 (2016).
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S. M. Barnett, L. Allen, R. P. Cameron, C. R. Gilson, M. J. Padgett, F. C. Speirits, and A. M. Yao, “On the natures of the spin and orbital parts of optical angular momentum,” J. Opt. 18, 064004 (2016).
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2015 (13)

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photonics 9, 822–826 (2015).
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D. A. B. Miller, “Perfect optics with imperfect components,” Optica 2, 747–750 (2015).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
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R. Solimene, C. Mola, G. Gennarelli, and F. Soldovieri, “On the singular spectrum of radiation operators in the non-reactive zone: the case of strip sources,” J. Opt. 17, 025605 (2015).
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R. Solimene, M. A. Maisto, and R. Pierri, “Inverse scattering in the presence of a reflecting plane,” J. Opt. 18, 025603 (2015).
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B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
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M. Franceschetti, “On Landau’s eigenvalue theorem and information cut-sets,” IEEE Trans. Inf. Theory 61, 5042–5051 (2015).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
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M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9, 529–535 (2015).
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S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, “Programmable multiport optical circuits in opaque scattering materials,” Opt. Express 23, 3102–3116 (2015).
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J. Yoon, K. Lee, J. Park, and Y. Park, “Measuring optical transmission matrices by wavefront shaping,” Opt. Express 23, 10158–10167 (2015).
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M. Kim, W. Choi, Y. Choi, C. Yoon, and W. Choi, “Transmission matrix of a scattering medium and its applications in biophotonics,” Opt. Express 23, 12648–12668 (2015).
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I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express 23, 12189–12206 (2015).
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2014 (6)

2013 (13)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space division multiplexing in optical fibers,” Nat. Photonics 7, 354–362 (2013).
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M. Mirhosseini, B. Rodenburg, M. Malik, and R. W. Boyd, “Free-space communication through turbulence: a comparison of plane-wave and orbital-angular-momentum encodings,” J. Mod. Opt. 61, 43–48 (2013).
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D. A. B. Miller, “Establishing optimal wave communication channels automatically,” J. Lightwave Technol. 31, 3987–3994 (2013).
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D. A. B. Miller, “Self-aligning universal beam coupler,” Opt. Express 21, 6360–6370 (2013).
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D. A. B. Miller, “Self-configuring universal linear optical component,” Photon. Res. 1, 1–15 (2013).
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S. Kosmeier, A. C. De Luca, S. Zolotovskaya, A. Di Falco, K. Dholakia, and M. Mazilu, “Coherent control of plasmonic nanoantennas using optical eigenmodes,” Sci. Rep. 3, 1808 (2013).
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R. Solimene, M. A. Maisto, G. Romeo, and R. Pierri, “On the singular spectrum of the radiation operator for multiple and extended observation domains,” Int. J. Antennas Propag. 2013, 585238 (2013).
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R. Solimene, M. A. Maisto, and R. Pierri, “Role of diversity on the singular values of linear scattering operators: the case of strip objects,” J. Opt. Soc. Am. A 30, 2266–2272 (2013).
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J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for millimeter-wave communications: system architecture, modeling, analysis, and measurements,” IEEE Trans. Antennas Propag. 61, 3814–3827 (2013).
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A. Özgür, O. Lévêque, and D. Tse, “Spatial degrees of freedom of large distributed MIMO systems and wireless ad-hoc networks,” IEEE J. Sel. Areas Commun. 31, 202–214 (2013).
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D. A. B. Miller, “Reconfigurable add-drop multiplexer for spatial modes,” Opt. Express 21, 20220–20229 (2013).
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M. Tamagnone, C. Craeye, and J. Perruisseau-Carrier, “Comment on ‘Reply to Comment on’ Encoding many channels on the same frequency through radio vorticity: first experimental test’,” New J. Phys. 15, 078001 (2013).
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D. A. B. Miller, “How complicated must an optical component be?” J. Opt. Soc. Am. A 30, 238–251 (2013).
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2012 (12)

M. Tamagnone, C. Craeye, and J. Perruisseau-Carrier, “Comment on ‘Encoding many channels on the same frequency through radio vorticity: first experimental test’,” New J. Phys. 14, 118001 (2012).
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A. Karoui and I. Mehrzi, “Asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions,” Appl. Math. Comput. 218, 10871–10888 (2012).
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J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14, 083001 (2012).
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K. Piché, J. Leach, A. S. Johnson, J. Z. Salvail, M. I. Kolobov, and R. W. Boyd, “Experimental realization of optical eigenmode super-resolution,” Opt. Express 20, 26424–26433 (2012).
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R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
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A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
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D. B. Conkey, A. N. Brown, A. M. Caravaca-Aguirre, and R. Piestun, “Genetic algorithm optimization for focusing through turbid media in noisy environments,” Opt. Express 20, 4840–4849 (2012).
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T. Cizmar and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
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I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20, 10583–10590 (2012).
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Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012).
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R. N. Mahalati, D. Askarov, J. P. Wilde, and J. M. Kahn, “Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling,” Opt. Express 20, 14321–14337 (2012).
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D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express 20, 23985–23993 (2012).
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2011 (5)

2010 (4)

J.-F. Morizur, L. Nicholls, P. Jian, S. Armstrong, N. Treps, B. Hage, M. Hsu, W. Bowen, J. Janousek, and H.-A. Bachor, “Programmable unitary spatial mode manipulation,” J. Opt. Soc. Am. A 27, 2524–2531 (2010).
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P. Martinsson, H. Lajunen, P. Ma, and A. T. Friberg, “Communication modes in vector diffraction,” Optik 121, 2087–2093 (2010).
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S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
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R. Somaraju and J. Trumpf, “Degrees of freedom of a communication channel: using DOF singular values,” IEEE Trans. Inf. Theory 56, 1560–1573 (2010).
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2009 (2)

M. Franceschetti, M. D. Migliore, and P. Minero, “The capacity of wireless networks: information-theoretic and physical limits,” IEEE Trans. Inf. Theory 55, 3413–3424 (2009).
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A. Karoui and I. Mehrzi, “Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions,” J. Comp. Appl. Math. 233, 315–333 (2009).
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2008 (6)

M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
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Specifically, I presume basic algebra and calculus, elementary real analysis (including convergence and limits), basic linear algebra including matrices with eigenvectors and eigenvalues, differential equations including elementary partial differential equations and eigenfunctions and eigenvalues, integral equations at least up to elementary Green’s functions, vectors, elementary vector calculus, basic notions of sets including the usual sets of numbers (integer, real, complex), basic wave equations, and electromagnetism up to and including Maxwell’s equations in differential form. Such mathematics is covered well by a text such as [127]. I specifically do not presume any knowledge of functional analysis (which would be relatively uncommon for physical scientists and engineers), and I do not require advanced knowledge of electromagnetism, such as the use of vector potentials and gauges. The advanced electromagnetism and the functional analysis needed are covered later in this work and in Ref. [122].

The various mathematical properties of finite matrix eigenproblems can, however, be deduced from the results of functional analysis; a finite matrix is then a special case.

D. A. B. Miller, “An introduction to functional analysis for science and engineering,” arXiv:1904.02539 (2019).

The pulses in a general time-dependent field would have to be square-integrable, but physically that essentially corresponds to finite energy, which we would expect anyway.

Note that we need make no distinction between spontaneous and stimulated photons because we are only considering one mode at a time here, and there is indeed anyway no distinction between stimulated and spontaneous photons in a given mode.

A first problem with the Coulomb gauge is that the equation for the scalar potential is unphysical in that any change in charge density in any region of space results in instantaneous changes of the potential ΦC everywhere in space. This apparent inconsistence with relativity does not result in actual violations of the velocity of light propagation of the fields E and B [214,248], but it is at least awkward. A second problem with the Coulomb gauge is that in wave problems we typically proceed by separation into “longitudinal” and “transverse” current densities. In free space, with no actual current densities anywhere in space, this causes no additional problems, but if there is indeed any current density at any point or finite region in space (and we expect to have source densities in our problems), the resulting longitudinal and transverse effective source current densities actually extend through all space [244].

Note, incidentally, that, though we are just using the “exp(−iωt)”AωMj(rR) parts in performing this calculation of energy, the resulting energy is the energy of the total real field AR(rR,t) because of the way we set up the energy inner products.

We have drawn this with grating couplers in vertical lines at the ends of the waveguides, but this is just an example. We can have any optics between the “source” waveguides on the left and the “receiver” waveguides on the right, and in two-dimensional arrangements, not just these vertical lines.

For this particular approach to work, the optical system has to be reciprocal.

In my opinion, [206] is incorrect in every substantial criticism made of my response [23] to those authors’ earlier paper on acoustic “orbital” angular momentum beams [22]. I used the term “optical angular momentum,” which is one of the terms in the field (see [31]), and I have not confused acoustic and optical communication. In my opinion, my paper [23] stands correct as written. See [208] for specific comments.

I have calculated with the scalar Green’s function, which is a first approximation in optics, but is the right approach for these acoustic waves, and it is acoustic channels that I calculated. With my approach, using communications modes, I achieved more channels, with fewer transmitters and receivers, and, contrary to these authors’ statements, my approach has no crosstalk in principle, not the −7.7  dB asserted by these authors (the −7.7  dB refers to channel strengths, not crosstalk).

Just to construct an orthogonal basis set AωMbn(rR) in VR, it is not strictly necessary to choose subsequent Jωbn(rS) to be orthogonal to all preceding ones Jωbm(rS); linear independence would be sufficient to allow construction of orthogonal AωMbn(rR). But choosing the Jωbn(rS) to be mutually orthogonal means that in this process we also usefully generate an orthogonal basis for the source functions.

Note that there are few restrictions on what form these current sources take—they are not functions that have to obey Maxwell’s equations, for example. Overall, we would require conservation of charge, but that will be automatic if these are monochromatic, and therefore purely oscillatory, functions. These orthogonal current sources could be as simple as small uniform patches on a surface. We would of course have to choose vector directions for such current patches, and if we want the resulting set to be complete, we should include versions with the currents in three vector directions that are at right angles.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Oxford University, 1995).

H. Haken, Light (North-Holland, 1981), Vol. 1.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).

Using a notation after [198], the nth such eigenfunction solution in the y direction in a given source or receiver aperture is of the form S0n(cL,ky/L), where the parameter cL=(π/2)NPy, with NPy being the paraxial heuristic number in the y direction. A similar set of solutions will exist in the x direction.

If we try to derive solutions with transverse boundaries at infinity using integral equations (as in our Green’s function SVD approaches), the corresponding singular values are all identical (there is no “loss” in the propagation because no wave “misses” the receiver space), so the eigensolutions are completely degenerate and any orthogonal basis is equally good, so the solution becomes mathematically trivial. Hermite–Gaussians will be solutions in one direction, for example, but so will any other set of complete functions. There is also a physical contradiction in such solutions without finite boundaries because we are violating the paraxial approximation by allowing the boundaries to extend arbitrarily in the transverse direction.

Beams with a specific “orbital” angular momentum correspond to a phase variation in azimuthal angle ϕ with an integer “quantum number” m in the form exp(imϕ). If the solutions with exp(imϕ) and exp(−imϕ) are degenerate, then we are free to construct the linear combinations with phase variations of the form cos(mϕ)=(1/2)[exp(imϕ)+exp(−imϕ)] and sin(mϕ)=(1/2i)[exp(imϕ)−exp(−imϕ)]. These new solutions, each being equal sums of solutions with equal but opposite “orbital” angular momentum, have zero “orbital” angular momentum. It is then a matter of taste whether we want to work with positive and negative m and exp(imϕ) solutions (with net “orbital” angular momentum), or positive m with cos(mϕ) and sin(mϕ) solutions (with no net “orbital” angular momentum). The total number of orthogonal functions available up to some specific |m| is exactly the same.

There is a small imaginary component left near the ends of the line of receiver points, so the wave is not exactly confocally curved there, though the real part is still quite a good representation of the overall wave amplitude there.

We are introducing the term “paraxial degeneracy” here.

Note that the sum rule S is different for each of these cases, and it would be wrong to conclude that the coupling strengths are generally reducing in magnitude as we make the volume thicker, even with non-uniform shapes. Generally, increasing the thickness (while correspondingly increasing the number of points in the volume) increases the absolute coupling strength. As we increase thickness non-uniformly, as in ellipsoidal source volumes, for some of the modes, the increase in coupling strength is more than for others.

If we increase the length of the line of receivers and correspondingly increase the separation between the sources and the receivers, so the angle subtended by the source line at the receivers is essentially constant, then the “knee” in the curves here moves closer to NHy—that is, the factor that here is 0.985 moves closer to 1. The form of the curve, explicitly including the exponential decay rate, does not change, however, with the singular values falling off exponentially with the same exponent.

In this rationalization, we presume that we can approximately “factorize” the modes into a product of “horizontal” and “vertical” mode forms, like those seen with “line” sources and receivers. Up to n≃NH, both the horizontal and vertical forms are for modes below the corresponding NHx and NHy limits. However, for n>NH, one or the other of the horizontal or vertical forms must exceed its corresponding NHx and NHy limits. So, there will be a set of NHx “horizontal” modes that correspond to the first “vertical” mode past the limit, and similarly a set of NHy modes that correspond to the first “horizontal” mode past the limit. So we expect to see a “step” with ≈NHx+NHy modes with approximately equal singular values. A similar argument for successive weaker modes in one or the other direction leads to a subsequent step, and so on. Because there is a number of such modes on each step that is therefore proportional (in this square case) to NHx=NHy=NH, we divide by NH in the exponential. Of course, this is not quite a complete counting of all the possible weakly coupled modes, because there will also be modes in which both the “horizontal” and “vertical” modes are both “weak,” so this rationalization is not a complete description, but it does give some sense as to why we can see “steps” and the NH factor in the denominator in the approximate exponential.

The general behavior of singular values for prolate spheroidal functions is well known [157] and expands this discussion for the weakly coupled values in a general “Fourier-transform” approach, showing that the number of “degrees of freedom” increases only logarithmically as the minimum acceptable singular value is decreased {[157], Eq. (2)}, which is consistent with an exponentially decaying strength of the singular values. Reference [158] extends this to more dimensions. Insofar as these Fourier-transform approaches are valid, which may hold in the limit of large structures separated by even larger distances, they give some explanation for this phenomenon.

C. A. Balanis, ed., Modern Antenna Handbook (Wiley, 2008).

We used a “spiral” approach to obtain an approximately uniform distribution of points on the spherical surface. See E. B. Saff and A. B. J. Kuijlaars, “Distributing many points on a sphere,” The Mathematical Intelligencer 19, 5–11 (1997), in our case using a “golden section” ratio angular increment π(3−5). A pseudo-code version of this algorithm is as follows for distributing Nsp points on a sphere of unit radius, returning arrays x, y, and z of the corresponding x, y, and z values for each point. Here, “;” is a statement separator, and xk is the kth element of the array x. g≔0; dz≔2/Nsp; s≔1−dz/2; dg≔π(3−5); For k≔1 to Nsp r≔1−s2;xk≔r cos(g); yk≔r sin(g);zk≔s; s≔s−dz; g≔g+dg.

The modal amplitudes used were chosen somewhat arbitrarily, but with suitable values for modes 12, 13, and 14, to illustrate the key points without also showing extreme behavior (such as very large source amplitudes at specific sources). The actual modal amplitudes in this example for the desired received field are (to three significant figures), in order for modes 1 to 14, 0.183, 0.143, 0.148, 0.346, 0.445, 0.207, 0.188, 0.395, 0.469, 0.198, 0.247, 0.104, 0.178, and 0.099. The phases of the modes were chosen randomly with a uniform distribution over all phases from −π to π; explicitly, those phases, for modes 1 to 14, are π times the following values: 0.823, 0.912, −0.668, −0.235, −0.141, −0.353, 0.678, 0.749, −0.919, 0.711, −0.098, −0.170, 0.935, and 0.325. (All numbers are quoted to three significant figures, though the actual values used had higher precision.)

G. de Villiers and E. R. Pike, The Limits of Resolution (Taylor and Francis, 2016).

G. Lerosey, F. Lemoult, and M. Fink, “Beating the diffraction limit with positive refraction: the resonant metalens approach,” in Plasmonics and Super-Resolution Imaging, Z. Liu, ed. (Pan Stanford, 2017), pp. 33–90.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, 1978).

D. Porter and D. S. G. Stirling, Integral Equations: A Practical Treatment, from Spectral Theory to Applications (Cambridge University, 1990).

J. K. Hunter and B. Nachtergaele, Applied Analysis (World Scientific, 2001).

In (IP2), for good reason, we choose a notation convention here that is the other way around from most (but not all) mathematics texts. Common mathematical notation for (IP2) would have (aγ,α)=a(γ,α), which, with (IP3), would give (γ,aα)=a*(γ,α). Our choice corresponds better with the order we encounter in our “algebraic shift” to Dirac notation, and gives a natural form of the associative property of multiplication as in matrix-vector notation.

This term “underlying inner product” is one that I am introducing here for clarity.

Note that this is technically a reuse of a notation; we already used (α,β) with such ordinary braces for the inner product. Such reuse is unfortunately rather common in mathematical texts.

Unfortunately, this “infinitely long” aspect of a given sequence may well not be stated clearly or explicitly in functional analysis.

In a metric space with a metric d(α,β), a sequence (αn) is said to be Cauchy (or to be a Cauchy sequence) if for every real number ε>0 (no matter how small) there is a number N (a positive integer or natural number) such that, for every m,n>N, d(αm,αn)<ε.

This use of “complete” in a “complete set” is different from the idea of a “complete” space; this confusion is unfortunate, but is unavoidable because of common usage.

Technically, the supremum is the smallest number that is greater than or equal to all the numbers being considered.

In a notation like this with this “dot”, it is best to view these inner-product operations as “waiting to happen”; just how much of the inner-product operation we are effectively writing down here can be somewhat vague in mathematics texts. However, we will take the approach that both any “operator weighting” and any integral for the inner product are “waiting to be applied” and in that sense are not yet part of this expression.

In its more common use in quantum mechanics, Dirac notation is not required to deal with the sophistication of different underlying inner products, though we see here that, with careful definitions, it can handle this extension.

Quite generally, a form such as |βj⟩2⟨αk|1 is an outer product. In contrast to the inner product, which produces a complex number, and which necessarily only involves vectors in the same Hilbert space, the outer product generates a matrix from the multiplication in “column-vector row-vector” order, and can involve vectors in different Hilbert spaces.

The same problem does not arise in finite-dimensional spaces; if we construct an infinitely long sequence made up from just the finite number of basis vectors in the space, we will have to repeat at least one of the basis vectors an infinite number of times, which gives us at least one convergent subsequence—the (sub)sequence consisting of just that basis vector repeated an infinite number of times. In fact, we can prove [122] that it is sufficient that an operator has finite-dimensional range for it to be compact. A corollary is that operators described by finite matrices are compact.

Note that there is some variation in notation in mathematics texts. Kreyszig [169] uses this definition for a positive operator, for example, and if the “≥” sign is replaced by a “>” sign in Eq. (133), he would then call the operator positive-definite. Others, however, such as [170], would give Eq. (133) as defining a non-negative operator, using “positive operator” only if the “≥” sign is replaced by a “>” sign.

Both “operator-weighted inner product” and “transformed inner product” are terms we are adding here; I know of no other standard names for these concepts.

Incidentally, note that, unlike many basis transformation, there is no requirement here that this transform is unitary.

In fact, it is not even necessary with such a “1/r2” integrand that the volumes do not overlap; the result of such an integral will be finite even if the resulting “1/r2” singularity is included. See the discussion in [187], p. 140 and p. 173.

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics (Springer, 2002).

Here, as noted in the discussion after Eq. (105), we have made the notational choice to leave the integrals over rR and rS out of this part of the mathematics, including them later when we perform the actual inner products.

C. Huygens, Traité de la Lumiere (Leyden, 1690). [English translation by S. P. Thompson, Treatise on Light (Macmillan, 1912).]

The full solution for scalar waves requires two kinds of sources, which can be written as point sources and spatial dipoles, but can also be written as spatio-temporal dipoles [145]; only one kind of such spatio-temporal dipole is typically required, however, for “slowly varying” wavefronts, allowing a return to a simple view of effective sources on the wavefront, and hence a simple Green’s function with an obliquity factor included.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994).

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

It is convenient algebraically to pretend that the time dependence of the wave is of the form exp(−iωt) and to work with complex amplitudes. Classical waves are, however, real, but we can always get back to that by adding the complex conjugate at the end.

We will introduce a somewhat more general definition of adjoint operators below, but for matrices, this definition is sufficient.

G. Strang, Linear Algebra and its Applications, 3rd ed. (Harcourt Brace Jovanovich, 1988).

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. (Cambridge University, 2013).

We postpone the definition of a compact operator since this is, unfortunately, rather technical. However, as we will argue below, the coupling operators associated with wave equations are Hilbert–Schmidt operators, which are all compact. The important results we discuss here are therefore going to apply very generally in wave problems.

To be quite technically correct here, if any of the eigenvalues is zero, then the set is not necessarily complete, but it can always be extended, for example, by a process such as Gram–Schmidt orthogonalization, to be complete.

If NS and NR are not equal, then all of the “extra” singular values formally associated with the larger of the two matrices are identically zero. That is, if Nlarge is the larger of NS and NR and Nsmall is the smaller of NS and NR, then all of the “extra” singular values sj from j=Nsmall+1 to j=Nlarge are zero. Formally, also, we are free to generate orthogonal eigenfunctions associated with these extra “0” singular values that are orthogonal to all of the first Nsmall eigenfunctions and to each other, by some process such as Gram–Schmidt orthogonalization. These “extra” eigenfunctions do not participate at all in communicating with waves between the sources and the receivers and, other than being orthogonal to one another and to the first Nsmall eigenfunctions, their choice is relatively arbitrary because they are also solutions of a degenerate eigenproblem, all sharing the same eigenvalue of zero.

At least, the sets are complete for discussing communications between the volumes, and they can be extended beyond that if necessary. See notes [133] and [134].

Note, incidentally, that such degeneracies are always finite for eigensolutions of compact Hermitian operators.

Note that, quite generally, S=Tr(GSR†GSR)=Tr(GSRGSR†). The trace of a matrix does not depend on the complete orthonormal basis set(s) used to represent it and is therefore also the sum of the eigenvalues of the matrix [because we could represent the matrix on the eigen (or SVD) basis set(s)].

Note that the eigenvectors of both the GSR†GSR and GSRGSR† operators are each arbitrary within a unit phase factor (e.g., of the form exp(iθ) for some angle θ). In practice, this may mean that we make some choice of phase for each of the eigenvectors that is convenient for us, and if or when we need the singular values sj rather than just the squared modulus |sj|2, we can formally establish the phase factors by computing, say, GSR|ψSj⟩≡sj|ϕRj⟩, as in Eq. (31).

Somewhat higher numerical precision is needed to see this orthogonality precisely.

Other configurations of the phase shifters in the block are possible and ultimately equivalent; for example, we could use one phase shifter on the top arm of the interferometer to control the split ratio and a second phase shifter on, say, the top output waveguide to control an additional phase. See [24–28].

For example, as discussed in [24] and [25], for an input signal in the top ESIn1 guide, we can progressively set the S11 and S22 blocks and the S31 phase shifter to give the desired power splitting and output phases for the first channel (i.e., the vector |ψS1⟩ of output amplitudes). For the second channel, with input in the middle ESIn2 guide, because we know the settings of the S11, S22, and S31 elements, we can readily calculate what outputs are required from the S12 and S22 elements to achieve the |ψS2⟩ set of output amplitudes at the right and hence we can calculate how to set those elements. We might think that we do not have enough elements to allow us to specify |ψS2⟩, but we do because it is guaranteed to be orthogonal to |ψS1⟩, which reduces the number of required independent parameters by two. For the final ESIn3 input, because |ψS3⟩ must be orthogonal to both |ψS2⟩ and |ψS1⟩, the only remaining independent parameter to set is the phase shift.

The half-wavelength spacing clarifies the behavior of the resulting waves because at spacings of half a wavelength or shorter, additional “diffraction orders” are eliminated, so there are no spurious additional beams to confuse the pictures of the waves.

The phase of the source and receiver modes is arbitrary, as is generally the case with such eigenmodes; multiplying an eigenmode by a complex factor still leaves it as a solution of the same (linear) eigenproblem. Furthermore, these eigenproblems only give us |sj|2, which similarly leaves us free to choose the phase of sj to be whatever is convenient.

Because these sources are actually in three-dimensional space, they are also transmitting in the directions in and out of the plane, and, indeed, actually equally well in all directions in the horizontal plane.

It is, incidentally, interesting to see how and why these pairs of sources work. Note, first, that in the calculations to generate the modes we made no prescription about the relative amplitudes and phases of the two different lines of sources. The resulting amplitudes and phases result entirely from the solution of the eigenvalues and eigenfunctions of the relevant matrix (GSRGSR† or GSR†GSR). In establishing the best possible source amplitudes, the numerical solution has “found” an approach that can be called a “spatiotemporal dipole” [145]. An ideal such spatiotemporal dipole would have equal and opposite amplitude for the two sources in the dipole (one on the “left” and one on the “right”) but with a phase lag on the “left” source that corresponds to the time taken for the wave to travel between the two sources in the pair. That leads to at least partially constructive addition on the “right” but destructive interference on the left. In this case, we see numerically that the amplitudes of the left and right sources in each pair are indeed approximately equal in magnitude, and the left source does indeed lag to the right by approximately the right phase [90° (π/2) for sources separated by a quarter wave]. Note again that the solution of this problem “found” this desirable behavior automatically; we did not “tell” the mathematics to find such spatiotemporal dipole solutions. Such spatiotemporal dipoles are also a particularly elegant way to restate Huygens’ principle [145], giving much better numerical results than the simple point sources of Huygens’ original proposal and eliminating unphysical backward waves.

So far, for simplicity, we presented SVD with equal numbers of source and receiver points, which resulted in a square matrix for GSR. In fact, though, such equal numbers are not necessary for SVD, and, correspondingly, SVD can be performed on a matrix that is not square. In our present case, though we have doubled the number of source points to NS=18, we can keep the number of receiving points at NR=9. In such a case the matrix GSR is a 9×18 matrix rather than a square one. In this case, the matrix GSR†GSR is an 18×18 matrix, whereas the matrix GSRGSR† is 9×9, which might seem to give a contradiction. Solving the GSR†GSR eigenproblem would give 18 eigenfunctions, whereas solving the GSRGSR† eigenproblem would give only nine. The resolution of this paradox is that the eigenvalues (and the singular values) for the additional nine eigenfunctions in the GSR†GSR case are mathematically identically zero [134]. The corresponding source functions have mathematically absolutely no coupling strength to the receivers. In our numerical calculations, the power coupling strengths of these additional modes are approximately 10−17 times as small rather than being exactly zero, with this finite but small value presumably reflecting rounding errors and limitations in the numerical calculations.

The mode in Fig. 7(a) is actually also the second most strongly coupled mode, though its coupling is smaller than the most strongly coupled mode only by a very small amount. The other two strongly coupled modes are analogous to those of Fig. 6(a) (a “two-bumped” mode) and Fig. 6(c) (a “three-bumped” mode), and these have very similar coupling strengths to one another in this case also. The percentages of the corresponding sum rule S for each of these three modes for the source and receiver arrangement of Fig. 7 are ∼28.04%, ∼28.51%, and ∼26.24%, for the “one-,” “two-,” and “three-” “bumped” modes, respectively.

Note, incidentally, that these power coupling strengths |sj|2 are not formal power coupling efficiencies between sources and receivers, nor are they necessarily even proportional to the power coupling efficiencies. We are not formally evaluating the total power emitted by the sources. These |sj|2 are the relative powers in each beam when starting with source functions of unit amplitude, but those unit amplitudes do not necessarily all correspond to unit emitted power.

B. Mamandipoor, A. Arbabian, and U. Madhow, “Geometry-constrained degrees of freedom analysis for imaging systems: monostatic and multistatic,” arXiv:1711.03585 (2018).

D. Tse and P. Viswanath, Fundamentals of Wireless Communication (Cambridge University, 2005).

Most of our use here will be for simple eigenvalue problems of the form M|ϕ⟩=a|ϕ⟩, but the mathematics can be extended to include generalized eigenvalue problems of the form M|ϕ⟩=aB|ϕ⟩, where B is also an operator.

D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University, 2008).

See [9], pp. 516–518.

When we want to describe a set of elements, such as the set of all functions |ψSj⟩, we can enclose the elements, or their description, inside “curly brackets,” as in {|ψSj⟩}. Generally, we should specify the range of an index such as j so that this set has more definite meaning, though we will typically omit this unless needed for clarity. (Often, it will be an infinite range, though usually we will implicitly start with j=1.)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).

A. E. Siegman, Lasers (University Science Books, 1986).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

U. S. Inan and A. S. Inan, Electromagnetic Waves (Prentice-Hall, 2000).

C.-N. Chuah, J. M. Kahn, and D. Tse, “Capacity of multi-antenna array systems in indoor wireless environment,” in GLOBECOM 1998 (IEEE, 1998), Vol. 4, pp. 1894–1899.

This statement with an amplitude Eoω(r) or Boω(r) is consistent with E(r,t)=Re(Eoω(r)exp[i(ωt+θe)]) or B(r,t)=Re(Boω(r)exp[i(ωt+θm)]). Such statements are common in electromagnetism textbooks (e.g., [242,243]).

Note too that θe and θm may vary with r.

The time average of cos2(ωt+θ) over a cycle is ½.

This form is also implicit for any particular frequency if the fields are Fourier-transformed and the frequencies in the Fourier transform are presumed to run over positive and negative values [241].

We could continue to write this same inner product as an operator-weighted inner product by multiplying out U¯¯=(U)†U (which gives a dyadic operator as a result, hence the notation). In that case, we could formally write the inner product at some time t as (μ,η)U¯¯≡(μ,U¯¯η)≡∫Vμ*(r,t)·U¯¯·η(r,t)d3r. This operator U¯¯ could be written as a 3×3 matrix operating to the right on the mathematical column vector of components of the vector potential field η, and on the left on the Hermitian adjoint of the mathematical vector of components of the vector potential field μ. However, that requires that we have the unusual situation of some derivatives operating to the left instead of to the right; that is mathematically straightforward, but it requires a correspondingly unusual notation, so for simplicity we omit it.

Note the similarity of this expression to the classical one in Eq. (189). Indeed, if we were to “symmetrize” aj*(t)aj(t)→(1/2)[aj(t)aj*(t)+aj*(t)aj(t)], rewrite the “a’s” as operators, and postulate the commutation relation Eq. (367), we would get Eq. (369).

One statement of the sampling theorem, due to Shannon (see [229]), is “using signals of bandwidth W one can transmit only 2WT independent numbers in time T.”

Though [150] was only finally published in 1961, it is the text of a 1951 lecture that had been distributed informally earlier.

We need to retain terms up to ∼ε2 to get a non-zero result in the algebra.

(|α⟩⟨β|)†=(⟨β|)†(|α⟩)†=|β⟩⟨α|.

There is a minor formal point that we have only proved these results for functions corresponding to non-zero eigenvalues (or singular values). For our communications problems, we can disregard any channels with zero coupling strength, so functions associated with zero singular values are of no interest. As far as the Hilbert spaces are concerned, we could either restrict them to using as basis sets only those eigenfunctions corresponding to non-zero singular values or extend the basis sets by a process such as Gram–Schmidt orthogonalization to construct basis sets for the larger spaces. None of those additional basis functions in either space will participate in the “communication” between the spaces, so it is of no consequence in our problem which of these formal approaches we use.

A unitary matrix B is one for which B†B=Iop, the identity operator. Quite generally, for any complete set {|γp⟩}, we can write Iop=∑p|γp⟩⟨γp| (such an expansion operating on any function in the space simply returns the same function). For our matrix here, we have U†U=(∑p|ψp⟩⟨γp|)†(∑q|ψq⟩⟨γq|)=∑p,q|γp⟩⟨ψp|ψq⟩⟨γq|=∑p|γp⟩⟨γp|=Iop, and similarly for V†V.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Pearson, 2013).

U. S. Inan, A. S. Inan, and R. K. Said, Engineering Electromagnetics and Waves, 2nd ed. (Pearson, 2015).

The subscript “C” is already in use for the Coulomb gauge, and the next letter (“o”) in the word “communications” has too many other uses, so we use the third letter, and continue in using uppercase letters for gauges, leading to the subscript “M” for this gauge (which also distinguishes it from the use of “m” for “magnetic,” as in Jm earlier).

The gauge in which the scalar potential is set completely to zero is known as the Hamiltonian or temporal gauge (see [149]). Here we retain a fixed scalar potential [as in Eq. (G30)] to deal with the static fields, which makes this M-gauge different from that Hamiltonian or temporal gauge.

It is possible with this M-gauge to write scalar wave equations for each of the vector components of AM, and driven by the corresponding vector components of an effective current density JM. However, the physical interpretation of this effective current density is somewhat involved and not particularly illuminating.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1994).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).

W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves (Morgan and Claypool, 2009).

Such behavior is, of course, also well described by tensors; a dyadic can in general just be viewed as a second rank tensor for a three-dimensional space and a dyadic can be written as a 3×3 matrix, with the three dimensions corresponding to three orthogonal unit vector directions.

Dyadic notation can also be viewed as an extension of vector and vector calculus notation, allowing obvious generalization of theorems and identities in vector and vector calculus algebra.

We presume that ∇δ(r−r′) is meaningful, which it will be if we approximate the delta function by an appropriate but very “sharp” function with continuous derivatives and formally take the limit as the function becomes “sharper.”

There is a very subtle point about Green’s functions for such a vector wave equation if we are looking at waves at points where there are also source current densities. In that case another “depolarization dyadic” term has to be added [257], equivalent to including an additional delta function term in the Green’s function itself. Since our sources are in one volume and the waves of interest to us are in another, we do not have to consider this term here, however.

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Figures (34)

Figure 1.
Figure 1. Conceptual view for (a) communications modes and (b) mode-converter basis sets. In both cases a source function | ψ S in a source or input volume V S , or more generally in a mathematical (Hilbert) space H S , results in a wave function | ϕ R in a receiving or output volume V R , or more generally in a mathematical (Hilbert) space H R . In the communications mode case (a) the coupling is through a Green’s function operator G S R as appropriate for the intervening medium between the spaces. In the mode-converter case (b), the coupling is through the action of a device (or scattering) operator D .
Figure 2.
Figure 2. Set of point sources at positions r S j in a source volume, and a set of point receivers at positions r R i in a receiving volume, coupled through the coupling operator G S R .
Figure 3.
Figure 3. Set of three sources, spaced by 2 λ in the “vertical” y direction, separated from three similarly spaced receiving points by a distance 5 λ in the “horizontal” z direction.
Figure 4.
Figure 4. Example electrical driving and receiving circuits to form the superposition of sources for transmission and to separate the channels again for reception. The input channels are the (voltage) bit streams V S In j , and the outputs are the corresponding (voltage) bit streams V R Out j . The feedback conductances d S and d R set the overall electrical gain in transmission and reception in the operational amplifier circuits that sum the currents on their input “bus” lines as required to form the necessary linear superpositions.
Figure 5.
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.
Figure 6.
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 | z | , 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.
Figure 7.
Figure 7. Two example modes using a double line of sources. (a) “Single-bumped” mode. (b) “Four-bumped” mode that is only approximately half as well connected as the first three modes. (As can be seen, large parts of it miss the receiver positions entirely.) (As in Fig. 6, the real part of the wave is plotted in false color, and the wave is multiplied by | z | for graphic clarity.) The use of a double line of sources avoids substantial “left-propagating” waves for these modes, making the beam behavior clearer in this larger picture. To avoid singularities in the graphics, the wave is not plotted in the region of the white rectangle. The two lines of sources are spaced by a quarter wave. In the plots of the source phase, the “left” column of sources lag the phase of the right column of sources by approximately 90° ( π / 2 ), and the amplitudes are approximately equal and opposite. [In (b), we have joined the amplitude points in a given vertical column of sources by dotted lines to guide the eye.] The fact that the sources within a given “left–right” pair have opposite amplitudes and are phase-delayed in this way comes out of the numerical solution, and is not a starting constraint. This behavior is typical of “spatiotemporal dipoles” [145], and the calculations have “found” these as the best sources here.
Figure 8.
Figure 8. (a) Layout of two-dimensional arrays of 17 × 17 = 289 point sources and receiver points, each on a square array, with points separated laterally by the wavelength λ , and the arrays separated, on the same axis, by 50 λ . (b) End view of the (real part of the) field amplitude for mode 1, with the receiver points superimposed. (c) In order, for the first 24 communications modes, the relative power coupling strengths ( | s j | 2 ), as a percentage of the power coupling strength sum rule S . (d) False color plots of the (real part of the) field amplitude at the receiver plane, together with the corresponding percentage of the sum rule. The dashed square represents the extent of the array of receiver points in each case, as in (b). Plots are relative to the maximum in each mode, and the false color amplitudes do not represent the relative coupling strengths of different modes.
Figure 9.
Figure 9. (a) Positions of sources and receivers superimposed on the beam intensity in the plane of the source and receiver points for a given mode. Here the intensity of the mode multiplied by the horizontal distance z from the source plane is shown in false color. This multiplication by z compensates in the graphics for an underlying fall-off in intensity proportional to 1 / z with such lines of sources. A small region immediately adjacent to the sources is not plotted so as to avoid singularities and/or some very large amplitudes there in the graphics. The sources consist of 97 pairs of sources in two vertical lines. The sources in a vertical line are spaced by λ / 2 ( λ is the wavelength), and the two lines of sources are spaced horizontally by λ / 4 (similarly to those in Fig. 7). (b) Histogram of the modulus squared of the singular values | s j | 2 for the different modes (numbered in decreasing order of the singular-value magnitude). These are shown as a percentage of the total sum rule S . (c) Relative magnitude of the singular values of each mode, compared to the first (and largest) singular value, and plotted on a logarithmic scale.
Figure 10.
Figure 10. Intensity graphs of the beams associated with the odd-numbered modes 1–19 for the source and receiver points, as in Fig. 9(a) (with intensities multiplied by the distance z from the sources for graphic clarity). Relative intensities are rescaled for graphic clarity for each mode plotted, so the absolute “brightness” has no meaning in comparing different modes.
Figure 11.
Figure 11. Illustration of two surfaces or phase fronts that are “confocally curved”—each one is curved around the center of the other surface: (a) from confocally curved surfaces; (b) using lenses with focal lengths f S and f R equal to the separation L .
Figure 12.
Figure 12. Graphs for the source function amplitudes (left column) and the receiver function amplitudes (right column) for modes 11, 13, and 19 for the source and receiver points as in Figs. 9 and 10. The points are the amplitudes, and the lines join adjacent points to guide the eye. Underlying approximately confocal phase curvatures have been removed in each case, and the set of points in each graph has been multiplied by a constant phase factor to make the resulting points approximately real for graphic clarity. Only the real parts of the source and receiver function amplitudes are plotted. The receiver amplitudes are for the normalized vector of amplitudes. The vector of source amplitudes is also normalized, but because only the amplitudes of the “right” of the two vertical lines of sources are plotted here, this vector is additionally multiplied here by 2 to give the source function amplitudes plotted here for clarity in comparison since only half of the sources are plotted. The | s j | are the magnitudes of the singular values for each mode.
Figure 13.
Figure 13. Intensity pattern at the receiving space for two point sources on the source space.
Figure 14.
Figure 14. Illustration of the solid angles (a)  Ω S subtended by the source surface (area A S ) at the receiving surface and (b)  Ω R by the receiving surface (area A R ) at the source surface.
Figure 15.
Figure 15. Coupling strengths | s n | 2 in decreasing order as a function of the mode number n as calculated for the SVD of the scalar wave coupling between the sets of source points and receiver points as shown. N H is the paraxial heuristic number, here A S A R / ( λ 2 L 2 ) , where A S and A R are the areas of the source and receiver surfaces, respectively, and L ( = 256 λ here) is the separation between the surfaces. Note that in each case | s n | 2 is nearly constant up to n = N H , after which it starts dropping rapidly.
Figure 16.
Figure 16. Paraxial cases with source or receiver spaces with shapes other than rectangles. (a) “L” shaped source space with a square receiving space. (b) Circular source space with a square receiving space. (c) and (d) Circular source and receiver spaces with (c) equal sizes and (d) different sizes. (All of these simulations were performed with arrays of source points and receiver points each spaced on a square grid of 2 λ in both the x and y directions and truncated to fit within the corresponding shapes.)
Figure 17.
Figure 17. Illustration of how non-uniform depth removes paraxial degeneracy. The receiving volume in all cases is a set of receiving points, on 2 λ centers in both directions that fit with a circle of radius 16 λ . For the red points and line, the source is an identical circle of source points, at a distance 128 λ away along their common axis. For the other sets of points and/or lines, the separation of the “faces” and the circular cross section are retained, but the source consists of the points (also spaced on 2 λ centers, now in all three directions) lying within a volume. For the gray dashed line, the volume is a cylinder of depth 8 λ . For the other three sets of points and lines, the bounding volume is half of an ellipsoid of revolution. ρ E is the ratio of the depth of the half ellipsoid compared to its cross-sectional radius (which is fixed at 16 λ ) (so ρ E = 1 would be a hemisphere). (a) shows the cross sections of the source points and the receiving points in each case. The inset in (b) shows a perspective view, with the circular cross sections and the axis of rotation indicated, of the ρ E = 1.5 case, which shows the ellipsoidal shape. The traces in (b) show the strengths of the various communications modes as a percentage of the sum rule.
Figure 18.
Figure 18. Plots of the relative size of the singular values for several different approximately paraxial pairs of lines of source and receiver points, as a function of the mode number n compared to the paraxial heuristic number N H y for each pair of source and receiver lines, on both linear (left graph) and logarithmic (right graph) scales. The values are shown as points, with the solid lines in the linear graph joining them to aid the eye. The dashed lines are exponential functions given in the text. Note that, because the fall-off of the singular values is very rapid above N H y , the horizontal scale is expanded to show the behavior just around N H y . Three cases are plotted for different lengths of source ( w s ) and receiver ( w r ) lines and separation ( L ), as sketched on the left. Red points and lines (upper traces): w s = w r = 1024 λ , L = 4096 λ , N H y 256 . Blue points and lines (middle traces): w s = w r = 2048 λ , L = 8192 λ , N H y 512 . Orange points and lines (lower traces): w s = w r = 4096 λ , L = 16384 λ , N H y 1024 . The number of source and receiver points used in the calculations was 513 (red), 1025 (blue), and 1036 (orange).
Figure 19.
Figure 19. Plots of the relative size of the singular values for several different approximately paraxial square arrays of source and receiver points, as a function of the mode number n compared to the paraxial heuristic number N H for each pair of source and receiver squares, on both linear (left graph) and a logarithmic (right graph) scales. The values are shown as points, with the solid lines in the linear graph joining them to aid the eye. The dashed lines are exponential functions given in the text. Note that the horizontal scale is displaced to show the behavior near to and above N H . The square source and receiver spaces have linear dimensions w s and w r , respectively, and separation L , and the points are equally spaced on square lattices in each case, and with an equal number of source and receiver points N . Red points and lines (upper traces): w s = w r = 40 λ , L = 160 λ , N = 441 , N H = 100 . Blue points and lines (middle traces): w s = w r = 60 λ , L = 240 λ , N = 961 , N H = 225 . Orange points and lines (lower traces): w s = w r = 80 λ , L = 320 λ , N = 1681 , N H = 400 . Gray points (right graph only): w s = 40 λ , w r = 80 λ , L = 640 λ , N = 1681 , N H = 400 .
Figure 20.
Figure 20. Illustration of the beam resulting from finding the best-coupled mode between two horizontal lines of sources and receivers, showing the longitudinal heuristic angle θ L . Both sources and receivers use 201 points spaced by λ / 4 , aligned in the z horizontal axis, and with center-to-center spacing of z o = 250 λ . From Eq. (71), θ L 141 mrad 8.1 ° and the corresponding δ y 35.35 λ . (a) is a cross section of the intensity. For graphic clarity, the magnitude is multiplied by z 2 once we leave the source region (technically, a factor [ max ( 30 λ , z ) ] 2 ). The intensity in the region immediately around the source is omitted from the graphics to avoid singularities. (b)  x y cross section of the intensity in the middle of the receivers; (c) perspective surface-plot view of the same data as in (b).
Figure 21.
Figure 21. Behavior of the magnitude of the singular values | s n | , as a function of communication mode number n , relative to that of the largest singular value | s 1 | , for three different centered spherical “shell” source and receiver spaces. In each case, the receiving points are on the surface of a 24 λ diameter sphere. The source points are on the surfaces of spheres of diameters 2 λ (upper, red line), 4 λ (middle, blue line), and 8 λ (lower, orange line), respectively. The solid lines are drawn between the calculated values of | s n | / | s 1 | in each case to guide the eye. The dashed and dotted lines are heuristic functions shown for comparison (see text). On the horizontal axis, the mode numbers for each curve are divided by the corresponding spherical heuristic numbers, which are N S H 2 50.3 , N S H 4 201 , and N S H 8 804 for the 2, 4, and 8 λ diameter source spheres, respectively. 1600 source and receiver points are used for the 2 λ and 4 λ cases, and 2400 for the 8 λ case, distributed approximately uniformly over the sphere surfaces [160].
Figure 22.
Figure 22. Example of constructing the required sources to generate a specific received “wave” or set of received amplitudes. To avoid the additional graphic complication of handling phases, the values plotted in the graphs are for the squared magnitudes of the relevant quantities. The source and receiver points are as in Fig. 9, with the corresponding modes as in Figs. 10 and 12. (a) Desired receiver values (points) and the calculated actual values generated (line) using the calculated source values. (b) Corresponding source values (points, joined by lines for visual clarity). (c) Values for each receiver mode used to construct the desired values at the receiver points. (d) Corresponding required values for each source mode to generate the desired receiver values. Note in particular in (c) and (d) that, though the required source values for each mode largely track the required receiver values for each mode for the first 10 modes (which all have similar singular values), for modes 12, 13, and 14 in particular [which are highlighted in (c) and (d)], the required values for the source modes have to rise because the singular values are becoming smaller.
Figure 23.
Figure 23. Plots of the required sources (the red points, joined by red lines for visual clarity, are the modulus squared of the amplitudes of the “front” line of sources) in the graphs on the left to attempt to synthesize the desired receiver values, shown as the red points on the graphs in the right (the modulus squared of the desired receiver amplitude is plotted). The actual resulting values of the modulus squared of the receiver amplitude are shown as the black line in these graphs on the right. Beam intensity, multiplied by the distance from the sources on the left for graphic clarity, is shown in the middle pictures in false color. Results for three different desired Gaussian widths are shown in (a), (b), and (c), respectively, with calculations based on using the first 20 communications modes. For (d) and (e), the calculation is restricted to using only 12 modes, and for (e), the desired position is shifted down by 18 λ .
Figure 24.
Figure 24. Illustration of the effect of changing the number of modes used in trying to create the Gaussian amplitudes for the six-wavelength-wide Gaussian shape in Fig. 23(c) [red points in (a) here]. The red bars in (b) show the mode amplitudes (modulus squared) required for the receiver communications modes (up to mode 23) to attempt to create the desired Gaussian shape. The gray bars show the corresponding relative strengths of the (modulus squared of) the source mode amplitudes up to mode 12. (Only odd-numbered modes occur in this problem because of the symmetry, and only those amplitudes are therefore plotted here.) The pink, blue, and orange colored bars show progressively the additional required (modulus squared) amplitudes for 16, 20, and 23 mode calculations. [The overall vertical position of the source mode (modulus squared) amplitudes on this logarithmic scale is adjusted to match the corresponding receiver mode (modulus squared) amplitude for the strongest coupled mode for easier comparison of relative magnitudes.] (a) shows that a sharper peak and a slightly narrow shape do result from adding further modes, but (b) shows that the required amplitudes of the additional higher-numbered modes become enormous, illustrating the practical impossiblity of substantially exceeding diffraction limits.
Figure 25.
Figure 25. (a) Receiver amplitude (modulus squared) for the desired “top-hat” function of width 18 λ (red points and dashed light red lines to guide the eye) and the various actual amplitudes (modulus squared) for use of different numbers of modes, similarly to Fig. 24(a). (b) Relative amplitudes of the (modulus squared of) the receiver and transmitter modes, similarly to Fig. 24(b).
Figure 26.
Figure 26. Illustration of current density elements J 1 and J 2 at r , in two directions e ^ 1 and e ^ 2 , both perpendicular to each other and to the vector R = r r , generating corresponding vector potential components A 1 and A 2 in those same directions at r . Note that, for propagating waves, a current density J R in the direction of R does not generate a corresponding “longitudinal” vector potential component in that direction at r , though near-field (“non-propagating”) terms can do so.
Figure 27.
Figure 27. Mode coupling strengths | s n | 2 and vector source current and (vector potential) fields for two example modes for an electromagnetic system with lines of point vector sources and receivers, spaced vertically by λ / 2 in lines 5 λ apart, as in the earlier scalar wave example of Fig. 6. Mode 4 is a well-coupled mode whose polarization, in both the current sources and the resulting waves, is in the plane of the paper and is also substantially tranverse to the propagation direction from left to right. The vector field plots and the source current vectors show only the real part of the complex values, and so are essentially “snapshots” of the current and field vectors. Some of the modes have source and wave polarization entirely in the direction out of the plane, and transverse to the propagation direction in this plane (red bars). All other modes have polarizations of currents and waves in the plane. Some of those are substantially transverse (blue bars); others of those are mixed between transverse and longitudinal (i.e., with polarization in the horizontal z direction) (gray bars). Some are almost entirely longitudinal (orange bars), with mode 14 being the strongest of these longitudinal modes. For graphic clarity, the amplitude of the wave is multiplied by a factor depending on the horizontal distance z from the sources. For mode 4, this factor is z to compensate for the expansion of the wave in the directions out of the plane. For mode 14, we also need to compensate for the additional " 1 / R " fall-off because the longitudinal wave Green’s function falls off at least as fast as 1 / R 2 . To prevent the particularly large near-field amplitudes from dominating the drawn vector lengths, the factor used is ( z 0.2 λ ) 3 / 2 rather than just z 3 / 2 . Though the amplitude changes from left to right are therefore artificial, the relative amplitudes within a vertical column of vectors, and the directions of the vectors, are, however, correct.
Figure 28.
Figure 28. Conceptual apparatus for finding the best orthogonal channels through any reciprocal linear scatterer or optical system at a given frequency (after [12]), nominally described here by some coupling operator D from left to right, using two interferometer meshes on either side of the scatterer. To find the most strongly coupled channel, we shine light into the “red” input waveguide 1 on the left, and adjust the interferometers in row A on the right to maximize the output “red” power in the output waveguide 1 on the right. Then we run in reverse, shining the “orange” power (actually, at the same wavelength as the “red” power—colors here are for graphic clarity only) backwards into the “output” waveguide 1 on the right, and adjust the row A interferometers on the left to maximize the “orange” power backwards out of “input” waveguide 1 on the left. We repeat this “red”/”orange” process forwards and backwards until the system converges, having found the most strongly coupled channel through the system. Then, leaving row A on both sides set, we repeat a similar process with the “green” and “purple” beams, now in waveguide 2 on both sides. This will find the second most strongly coupled channel. We can then repeat for the waveguides 3. (No final “waveguides 4” process is required, because it is automatically configured as the only remaining orthogonal channel.) The process has found the four most strongly coupled channels in this system. Technically, this process has effectively found the singular-value decomposition of the optical system between the waveguide amplitudes at the “source” dashed line on the left and those at the “receiving” dashed line on the right, effectively embedding the unitary matrices U and V of the SVD of D = VD diag U in the interferometer settings in the meshes on the left and the right. Adapted with permission from IEEE.
Figure 29.
Figure 29. Configurations to consider either the full coupling operator D from source to receiving volumes, or a succession of three operators, including an “internal” scattering operator D D D that generates effective sources for waves to the receiving volume from the incoming waves from the source volume.
Figure 30.
Figure 30. Singular-value decomposition architecture for constructing any matrix (here 3 × 3 ) in optics between a set of waveguide input and output amplitudes [25].
Figure 31.
Figure 31. Thought-experiment apparatus for establishing a modal thermal radiation law (after [7]).
Figure 32.
Figure 32. Schematic for estimating the largest size of source “patch” for which a uniform source density across the patch can be approximated by a point source in the middle of the patch.
Figure 33.
Figure 33. Two point sources A and B spaced 2 Δ z apart along the z axis at a very large distance z o from a point P . Point Q is spaced a relatively small distance δ y away from P in the y direction. θ is the angle subtended by the line segment PQ relative to the midpoint between A and B. (Not to scale; z o 2 Δ z .)
Figure 34.
Figure 34. Construction for deducing an effective number of degrees of freedom from a spherical source.

Tables (1)

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Table 1. Mode Coupling Strengths for Nine Point Sources and Receivers

Equations (392)

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A mode is an eigenfunction of an eigenproblem describing a physical system .
2 ϕ ω R ( r ) + k 2 ϕ ω R ( r ) = ψ ω S ( r ) ,
k 2 = ω 2 / v 2 .
G ω ( r ; r ) = 1 4 π exp ( i k | r r | ) | r r | .
ϕ ω R ( r ) = V S G ω ( r ; r ) ψ ω S ( r ) d 3 r .
ϕ ω R ( r Ri ) = 1 4 π j = 1 N S exp ( i k | r R i r S j | ) | r R i r S j | h j = j = 1 N S g i j h j ,
g i j = 1 4 π exp ( i k | r R i r S j | ) | r R r S j | .
f i = j = 1 N S g i j h j .
| ψ S = [ h 1 h 2 h N S ] , | ϕ R = [ f 1 f 2 f N R ] , and G S R = [ g 11 g 12 g 1 N S g 21 g 22 g 2 N S g N R 1 g N R 2 g N R N S ] ,
| ϕ R = G S R | ψ S .
( | ψ S ) [ h 1 h 2 h N S ] [ h 1 * h 2 * h N S * ] ψ S | ,
G S R [ g 11 g 12 g 1 N S g 21 g 22 g 2 N S g N R 1 g N R 2 g N R N S ] [ g 11 * g 21 * g N R 1 * g 12 * g 22 * g N R 2 * g 1 N L * g 2 N S * g N R N S * ] .
( GH ) = H G
( G | ψ ) = ψ | G .
( G ) = G
[ ( | ϕ ) ] = [ ϕ | ] = | ϕ ,
P i = f i * f i .
P = i = 1 N f i * f i = ϕ R | ϕ R = ( ψ S | G S R ) ( G S R | ψ S ) = ψ S | G S R G S R | ψ S ,
α | β α | | β .
| ϕ = q | ϕ q ,
ϕ p | ϕ q = 0 if and only if p q ,
P = ϕ | ϕ = ( p ϕ p | ) ( q | ϕ q ) = p , q ϕ p | ϕ q = q ϕ q | ϕ q = q P q ,
P q = ϕ q | ϕ q
ϕ R | ϕ R = [ f 1 * f 2 * f N R * ] [ f 1 f 2 f N R ] = j = 1 N R f j * f j = j = 1 N R | f j | 2 > 0 ,
ψ S j | ψ S j = 1 .
ψ S p | ψ S q = δ p q ,
δ p q = [ 1 if p = q 0 if p q .
( G S R G S R ) = G S R ( G S R ) = G S R G S R ,
G S R G S R | ψ S j = | s j | 2 | ψ S j .
G S R G S R | ϕ R j = | s j | 2 | ϕ R j .
G S R | ψ S j = s j | ϕ R j
G S R | ϕ R j = s j * | ψ S j .
G S R = j = 1 N m s j | ϕ R j ψ S j | ,
G S R = j = 1 N m s j * | ψ S j ϕ R j | .
G S R = V D diag U ,
S = q = 1 N m | s q | 2 = i = 1 N R j = 1 N S | g i j | 2 .
| s 1 | 2 | s 2 | 2 | s j | 2 .
the number of orthogonal channels  ( communications modes ) with power coupling strength | s | 2 | s j | 2 is j .
| r R 1 r S 3 | = ( y R 1 y S 3 ) 2 + ( z R 1 z S 3 ) 2 = λ 4 2 + 5 2 = 41 λ ,
g 13 = 1 4 π exp ( i k | r R 1 r S 3 | ) | r R 1 r S 3 | = 1 4 π exp ( 2 π i 41 ) 41 λ 0.01020 0.00711 i λ .
g S R = 4 π L z = 62.83 .
g S R g 13 62.83 × ( 0.01020 0.00711 i ) 0.64 + 0.45 i .
g S R G S R [ 1 0.7 + 0.6 i 0.64 + 0.45 i 0.7 + 0.6 i 1 0.7 + 0.6 i 0.64 + 0.45 i 0.7 + 0.6 i 1 ] .
S = 7.67 / g S R 2 .
g S R 2 G S R G S R [ 2.47 0.67 0.08 i 0.42 0.67 + 0.08 i 2.72 0.67 + 0.08 i 0.42 0.67 0.08 i 2.47 ] .
| s 1 | 2 = 3.41 g S R 2 , | s 2 | 2 = 2.89 g S R 2 , and | s 3 | 2 = 1.37 g S R 2 .
| s 1 | 2 + | s 2 | 2 + | s 3 | 2 3.41 g S R 2 + 2.89 g S R 2 + 1.37 g S R 2 = 7.67 g S R 2 = S .
| ψ S 1 = [ 0.41 0.81 + 0.1 i 0.41 ] , | ψ S 2 = [ 0.71 0 0.71 ] , and | ψ S 3 = [ 0.58 0.57 0.07 i 0.58 ] ,
| ϕ R 1 [ 0.41 0.81 0.1 i 0.41 ] , | ϕ R 2 [ 0.71 0 0.71 ] , and | ϕ R 3 [ 0.58 0.57 + 0.07 i 0.58 ] .
| ψ S q = [ a 1 q exp ( i θ 1 q ) a 2 q exp ( i θ 2 q ) a 3 q exp ( i θ 3 q ) ] and | ϕ R q [ b 1 q exp ( i η 1 q ) b 2 q exp ( i η 2 q ) b 3 q exp ( i η 3 q ) ] ,
S = 72.65 / g S R 2 .
| ψ S j [ h 1 j h 2 j h N S j ] T
ϕ j ( r ) = 1 4 π q = 1 N S exp ( i k | r r S q | ) | r r S q | h q j .
c S ( x S , y S ) = exp [ i k ( x S 2 + y S 2 + L 2 L ) ]
c R ( x R , y R ) = exp [ i k ( x R 2 + y R 2 + L 2 L ) ] ,
ϕ ( y ) exp ( i k ( y d s 2 ) + L 2 ) + exp ( i k ( y + d s 2 ) + L 2 ) ,
| ϕ ( y ) | 2 cos 2 ( π d s y λ L ) .
d r = λ L d s .
N H y = ( 2 Δ y S ) ( 2 Δ y R ) λ L .
N H y = 48 × 48 192 = 12 .
N H x = ( 2 Δ x S ) ( 2 Δ x R ) λ L .
N H = N H x N H y .
A S = ( 2 Δ x S ) × ( 2 Δ y S ) and A R = ( 2 Δ x R ) × ( 2 Δ y R ) .
N H = A S A R λ 2 L 2 .
Ω S A S / L 2 and Ω R A R / L 2 .
N H Ω S A R λ 2 Ω R A S λ 2 .
Ω S 1 = Ω S N H = λ 2 A R and Ω R 1 = Ω R N H = λ 2 A S ,
h ( n ) = 8 exp [ 0.811 ( n 0.985 N H y ) ] .
h ( r ) = 4 exp ( 3 5 ( n N H ) N H ) .
d p < λ L / 2 w .
θ L = λ 2 Δ z
Ω L A L z o 2 = π θ L 2 = π λ 2 Δ z ,
Ω S 1 Ω L 2 Δ z λ π ( 2 Δ x ) ( 2 Δ y ) .
N S H = 16 π r 2 / λ 2 ,
f S H ( n ) = 1 3 exp ( 3 8 ( n N S H ) N S H ) .
| ϕ R o = j a j | ϕ R j ,
a j = ϕ R j | ϕ R o .
| ψ S o = j a j s j | ψ S j j 1 s j ϕ R j | ϕ R o | ψ S j .
ϕ ( y R j ) = c R ( 0 , y R j ) exp ( ( y R j y o ) 2 w 2 ) .
For all vectors α , β , and γ in a vector space, and all (complex) scalars a , we define an inner product ( α , β ) , which is a (complex) scalar, through the following properties : ( IP 1 ) ( γ , α + β ) ( γ , α ) + ( γ , β ) , ( IP 2 ) ( γ , a α ) = a ( γ , α ) [where a α is the vector or function in which all the values in the vector or function α are multiplied by the (complex) scalar a ] , ( IP 3 ) ( β , α ) = ( α , β ) * , ( IP 4 ) ( α , α ) 0 , with ( α , α ) = 0 if and only if α = 0 the zero vector) .
α = ( α , α ) .
d P ( α , β ) α β = ( α β , α β ) .
a non-zero element α of an inner-product space is said to be orthogonal to a non-zero element β of the same space if and only if ( α , β ) = 0 .
A Hilbert space is a complete inner-product space .
( α j , α k ) = δ j k .
γ = a 1 α 1 + a 2 α 2 + j a j α j .
( α j , γ ) = a j .
There is always a basis for a Hilbert space .
γ = [ a 1 a 2 ] | γ .
η = k r k α k and μ = k t k α k ,
r k = ( α k , η ) and t k = ( α k , μ )
( μ , η ) = p , q t p * r q ( α p , α q ) = p , q t p * r q δ p q = p , q t p * r q [ t 1 * , t 2 * , ] [ r 1 r 2 ] .
| η [ ( α 1 , η ) ( α 2 , η ) ] and | μ [ ( α 1 , μ ) ( α 2 , μ ) ] ,
( μ , η ) [ ( α 1 , μ ) * ( α 2 , μ ) * ] [ ( α 1 , η ) ( α 2 , η ) ] μ | | η μ | η ,
γ = A α .
O 1 A ( α + β ) = A α + A β , O 2 A ( c α ) = c A α .
A sup = sup α in D α 0 A α R α D < .
A α A α ,
σ = A 21 η .
( μ , σ ) 2 ( μ , A 21 η ) 2 .
a j k = ( β j , A 21 α k ) 2 .
η = k r k α k and μ = j t j β j ,
r k = ( α k , η ) 1 and t j = ( β j , μ ) 2 .
( μ , A 21 η ) 2 = j t j * ( β j , A 21 [ k r k α k ] ) 2 = j , k t j * r k ( β j , A 21 α k ) 2 = j , k t j * a j k r k .
A 21 j , k ( · , β j ) a j k ( α k , · ) ,
| η [ r 1 ( = ( α 1 , η ) 1 ) r 2 ( = ( α 2 , η ) 1 ) ] and | μ [ t 1 ( = ( β 1 , μ ) 2 ) t 2 ( = ( β 2 , μ ) 2 ) ] .
A 21 [ a 11 a 12 a 21 a 22 ] .
j , k t j * a j k r k μ | A 21 | η .
μ | A 21 | η μ | ( A 21 | η ) ( μ | A 21 ) | η .
A 21 j , k a j k | β j 2 α k | 1 .
2 μ | A 21 | η 1 2 μ | ( j , k a j k | β j 2 α k | 1 ) | η 1 = p t p * 2 β p | ( j , k a j k | β j 2 α k | 1 ) q r q | α q 1 = q t p * j , k δ p j a j k δ k q q r q = j , k t j * a j k r k
A 21 j , k a j k | β j α k | .
( μ , A η ) 2 = ( A μ , η ) 1 ,
a j k = ( β j , A α k ) 2 and b k j = ( α k , A β j ) 1 .
b k j = ( α k , A β j ) 1 = ( A β j , α k ) 1 * = ( β j , A α k ) 2 * = a j k * ,
( A ) = A .
The operator A ( from the normed space F to the normed space G ) is compact if and only if it maps every bounded sequence ( α m ) of vectors in F into a sequence in G that has a convergent subsequence .
d ( α j , α k ) ( α j α k , α j α k ) = ( α j , α j ) + ( α k , α k ) ( α k , α j ) ( α j , α k ) = 1 + 1 0 0 = 2 .
For a Hilbert space H 1 with an orthonormal basis { α 1 , α 2 , } and a bounded operator A that maps from vectors in H 1 to vectors in a Hilbert space H 2 , A is a Hilbert–Schmidt operator if and only if S = j A α j 2 < .
A H S = S j A α j 2 .
S A H S 2 = j , k | a k j | 2 = j α j | A A | α j = k β k | A A | β k Tr ( A A ) = Tr ( AA ) .
Hilbert–Schmidt operators are compact .
if A is a Hilbert–Schmidt operator, so also are A , A A and AA
( β , A γ ) = ( A β , γ ) .
A = A ,
a j k = a k j * .
for a Hilbert–Schmidt operator A (which is not necessarily Hermitian), the operators A A and AA are both compact and Hermitian .
A α = c α ,
For a compact Hermitian operator A mapping from a Hilbert space H onto itself, the set of eigenfunctions { β j } of A is complete for describing any vector ϕ that can be generated by the action of the operator on an arbitrary vector ψ in the space H , i.e., any vector ϕ = A ψ . If all the eigenvalues of A are non-zero, then the set { α j } will be complete for the Hilbert space H ; if not, then we can extend the set by Gram–Schmidt orthogonalization to form a complete set for H .
A = j = 1 r j β j ( β j , · )
A = j = 1 r j | β j β j | .
The eigenvectors β j of a compact Hermitian operator can be found by a progressive variational technique, finding the largest possible result for A β j where β j is constrained to be orthogonal to all the previous eigenvectors. This will also give a corresponding set of eigenvalues r j in descending order of their magnitude .
( β , C β ) 0.
any operator that can be written in the form C = B B , where B is a linear operator, is a positive operator .
any eigenvalues c of a positive operator are positive (non-negative), i.e., c 0 .
if A is a Hilbert–Schmidt operator, then the operators A A and AA [in addition to being Hermitian Hilbert–Schmidt (compact) operators] are positive operators, and therefore any eigenvalues of either of them are necessarily positive (technically , 0 ) .
( β , γ ) W ( β , W γ ) .
( β , γ ) W = ( β , W γ ) = ( W β , γ ) = ( γ , W β ) * ( γ , β ) W * .
( γ , α + β ) W ( γ , W ( α + β ) ) = ( γ , W α + W β ) = ( γ , W α ) + ( γ , W β ) = ( γ , α ) W + ( γ , β ) W ,
( γ , a α ) W ( γ , W a α ) = ( γ , a W α ) = a ( γ , W α ) a ( γ , α ) W .
( α , β ) w = w ( x ) α * ( x ) β ( x ) d x
W = B B ,
( β , γ ) W ( β , W γ ) = ( β , B B γ ) = ( B β , B γ ) ,
( β , γ ) T B ( B β , B γ ) ,
G ω ( r R ; r S ) = 1 4 π exp ( i k | r R r S | ) | r R r S | .
V S | G ω ( r R ; r S ) | 2 d 3 r S = 1 16 π 2 V S 1 | r R r S | 2 d 3 r S C = V S 16 π 2 r min 2 ,
S = V R V S | G ω ( r R ; r S ) | 2 d 3 r S d 3 r R V R V S 16 π 2 r min 2 .
S V s w 4 π as r a ,
( μ S , η S ) V S μ S * ( r S ) η S ( r S ) d 3 r S and ( μ R , η R ) V R μ R * ( r R ) η R ( r R ) d 3 r R ,
G ω ( r R ; r S ) p , q g p q ( · , α R p ) ( α S q , · ) p , q g p q α R p ( r R ) α S q * ( r S ) .
g i j = V S V R α R i * ( r R ) G ω ( r R ; r S ) α S j ( r S ) d 3 r R d 3 r S .
| G ω ( r R ; r S ) | 2 [ p , q g p q α R p ( r R ) α S q * ( r S ) ] [ m , n g m n α R m ( r R ) α S n * ( r S ) ] * = p , q , m , n g p q g m n * α R p ( r R ) α R m * ( r R ) α S q * ( r S ) α S n ( r S ) .
V S V R | G ω ( r R ; r S ) | 2 d 3 r R d 3 r S = p , q , m , n g p q g m n * V R α R p ( r R ) α R m * ( r m ) d 3 r R V S α S q * ( r S ) α S n ( r S ) d 3 r S = p , q , m , n g p q g m n * δ p m δ q n = p , q | g p q | 2 .
S = V S V R | G ω ( r R ; r S ) | 2 d 3 r R d 3 r S = p , q | g p q | 2 .
the scalar Green’s function G ω ( r R ; r S ) = 1 4 π exp ( i k | r R r S | ) | r R r S | operating from a finite source volume V S to a receiver volume V R , which is either finite or a spherical shell of arbitrarily large radius, is a Hilbert–Schmidt operator .
we can establish the sum rule S without solving the SVD problem for the eigenfunctions and eigenvalues .
any finite coupling operator D ( r R ; r S ) between finite volumes V S and V R is a Hilbert–Schmidt operator .
any finite coupling operator D ( r R ; r S ) between finite volumes V S and V R , and for which any finite functions in the associated Hilbert spaces lead to finite operator-weighted inner products , is a Hilbert–Schmidt operator with respect to those inner products .
S b = p = 1 N R q = 1 N S | b p q | 2
S = y R = h / 2 h / 2 y S = h / 2 h / 2 | G ω ( y R ; y S ) | 2 d y S d y R = 1 ( 4 π ) 2 y R = h / 2 h / 2 y S = h / 2 h / 2 1 L 2 + ( y S y R ) 2 d y S d y R 0.726 ( 4 π ) 2 .
| g i j | λ / 2 × λ / 2 1 4 π ( L 2 + ( y S j y R i ) 2 ) ,
S b = i = 1 9 j = 1 9 | g i j | 2 0.726 ( 4 π ) 2 ,
| r S r R | L ( 1 + ( x R x S ) 2 2 L + ( y R y S ) 2 2 L ) .
E M = A M t ,
B M = × A M .
× × A ω M k 2 A ω M = μ J ω
k 2 = ω 2 ε μ ω 2 / v 2
v = 1 / ε μ ,
J ( r , t ) = J ω ( r ) exp ( i ω t ) + c.c .
G ¯ ¯ ω M P ( r ; r ) = ( e ^ 1 e ^ 1 + e ^ 2 e ^ 2 ) G ω ( r ; r ) ,
A · e ^ a e ^ b · J ( A · e ^ a ) ( e ^ b · J ) A a J b ,
G ¯ ¯ ω M = G ¯ ¯ ω M P + G ¯ ¯ ω M N
G ¯ ¯ ω M N = [ 1 k R ( i 1 k R ) ( 2 R ^ R ^ e ^ 1 e ^ 1 e ^ 2 e ^ 2 ) ] G ω ( R ) ,
E ω M = i ω A ω M .
B ω M = ( 1 / i ω ) × E ω M .
( μ , η ) V μ * ( r ) · η ( r ) d 3 r ,
u = 1 2 ( ε E · E + 1 μ B · B ) .
( μ , η ) T U ( U μ , U η ) ,
U ω = [ i ω ε 0 0 0 i ω ε 0 0 0 i ω ε 0 1 μ x 3 1 μ x 2 1 μ x 3 0 1 μ x 1 1 μ x 2 1 μ