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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.

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.

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.

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.

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).

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

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

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.

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.

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).

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.

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.

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.

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.

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).

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.

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

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.)

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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.

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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).