Fredholm's alternative breaks the confinement of electromagnetic waves

In general, for physical systems that can be described by a particular class of integral equations arise a situation very similar to that in the solution of a system of algebraic linear equations. In the last case, we have two different sets of solutions depending on the homogeneity of the linear equations. If the equations are homogeneous they have solutions different to the trivial zero if the system's determinant is equal to zero, the mentioned condition also implies the existence of eigenvectors and eigenvalues. But if the system is inhomogeneous, the existence for a unique solution depends on the non-vanishing value of the system's determinant. In case of Fredholm's equations, we know that a similar rule exists. The rule appears as a discrimination principle called the Fredholm's alternative.^1–5 It says that if the Fredholm's determinant is zero, we have that the homogeneous version of the integral equation have non-trivial solutions, but if we want that the inhomogeneous integral equation has a solution, we must demand that the Fredholm's determinant be different to zero. Before we make of the Fredholm's alternative (FA) the center of our attention we recall that only recently we can talk about the connection between evanescent waves^6–12 and resonant solutions of the homogeneous Fredholm's equation.^13–18 But then we have a physical interpretation of the solutions we want to classify, that is if the integral equation is inhomogeneous we have a positive refraction index-like conditions but if the equation that describes the physical situation is homogeneous we then have negative refraction index-like conditions and the solutions can be called resonances. Although in the two different kinds of physical conditions there are travelling waves that allow communications, their frequencies are basically different because in the positive refraction index-like conditions, the corresponding frequency values to the resonant ones are really hidden or in a practical sense are confined to the so called near zone without the possibility to bring any information because of their exponentially decaying amplitudes.^9,10 The paper is organized as follows: In order to visualize the relation between the ever changing real conditions and the academic strict dichotomy, we will show the generalized Fredholm's equation frequency dependent with a source term in the next section (section II). Then in section III we show the relation between the homogeneous and inhomogeneous Fredholm's equations via the Fredholm's eigenvalue and we create a special Fredholm's equation with the purpose of analyzing the change to resonant conditions. In section IV we establish the conditions that trigger the change in the broadcasting regime and that cause the braking of the confinement for the electromagnetic waves; also we show in a recent experiment¹¹ how it is possible to manipulate the left-hand material conditions in a “plasma sandwich” and we can note the presence of the evanescent waves and the resonances. In section V we suggest that the Fredholm's eigenvalue can be selected as a phase factor and consider its general behavior, also obtaining some interesting consequences. The concluding remarks appear on section IV. On appendix  A we show how we can understand the generalized Fredholm's equation as an integral equation that has the form of the equation obtained in a convergent second step of successive approximations procedure. And in appendix  B we justify to suppose that the eigenvalue can be selected as a phase factor.

Now we write the equation we called a generalized inhomogeneous Fredholm's equation,¹⁰ showing explicitly the function we call for convenience Fredholm's eigenvalue or as we will see in appendix  B, properly the inverse of it:

In this equation we can see three elements whose behavior defines the kind of media because we have those specific mathematical conditions are strongly related to physical conditions. These elements are |$f^{m( \circ)} (\overline r,\omega)$|$fm(∘)(r¯,ω)$ that is the source term, υ(ω) that is the function we named Fredholm's eigenvalue for convenience and of course the kernel |${\sfb K}_n^{m( \circ)} (\omega ;\overline r,\overline {r}^\prime)$|$Knm(∘)(ω;r¯,r¯′)$⁠; we additionally suppose that |$f^{m( \circ)} (\overline r,\omega)$|$fm(∘)(r¯,ω)$ could have a more complicated structure than a simple source, that is, contains a small contribution to the integral term in equation (1) that vanishes when we are in the neighbor of a resonance. Properly, at this stage υ(ω) is then an analytic function of the frequency, |$f^m (\overline r,\omega)$|$fm(r¯,ω)$ are the travelling solutions of the vector Fredholm's equation. But if we want to take advantage of the resonant solutions of the homogeneous vector Fredholm's equation we can follow the recipe for quantum Gamow (resonant) states.^5,15,17 That is we deform the path of integration over the real axis of equation (1) to include the resonances on the fourth complex quadrant in the spectral representation. Then we have the new spectral representation of the Green's function as follows^2,5,10,14,15

Is clear that our base now is formed by three different classes of electromagnetic solutions: confined field, travelling waves and resonant waves. We can explain the role of each set in the following manner:

The continuum confined field can be expressed by linear combinations of only the discrete so called confined frequencies. Travelling waves now have two sources; the first is the common physical broadcasting; but the other is the transformation of the confined field into travelling waves. More precisely, under certain physical conditions in which the media behaves like a right-hand material the complete field is separated in two regions: the near and far regions and described mostly by the two first terms in equation (2) but if there are left-hand material conditions, the middle and the final terms dominates. Then, we have reached our first goal of extend our mathematical base to include the resonances |$w_e^m (\overline r _k ;\omega)$|$wem(r¯k;ω)$⁠.

In case of left-hand material conditions the electromagnetic field is described by the homogeneous Fredholm's equation:¹⁰ 

In this equation |$w_e^m (\overline r ;\omega)$|$wem(r¯;ω)$ are the resonances and η_e(ω) is the eigenvalue corresponding to the e resonance. The theory of homogeneous Fredholm's equations^1–4 establishes that the condition for the existence of solutions is that the first Fredholm's minor |$M^m \left( {\overline r,\overline {r_0 } ;\omega } \right)$|$Mmr¯,r0¯;ω$ complies:^1,3,5,15

So, if we multiply equation (1) by Δ(η, ω) and taking the difference with equation (4) we can write after making some rearrangements

Now we define for convenience the following functions:

And also

By using equations (6) and (7) we can write equation (5) as

Equation (8) is also a generalized inhomogeneous Fredholm's equation (if the absolute value of the factor Δ(η, ω)[η(ω) − υ(ω)] can be considered very smaller than one) whose solutions can be explained (due to his definition) as a mix of two types of fields having in principle different behaviors, that is, the precursors to resonances (but not the properly resonances) and the traditional travelling ones. Meanwhile we have not resonance conditions we suppose that η(ω), υ(ω) and Δ(η, ω) have not particular values so equation (8) must describe a general behavior of the entire field, and the integral term in definition (7) could be a difficult task for solving equation (8) analytically. But if we go near the resonance conditions the source term vanishes and we recover equation (3) that is, the equation accomplished by the resonances. Now, if equation (4) has a physical interpretation rather than a purely mathematical, we must have that at least for a region in the complex plane of ω, the functions η(ω), and υ(ω) must coincide in order to have physical non resonant solutions of equation (4) coincident with solutions of equation (1) and we can go continuously from one to the other. In this region the third term in equation (7) vanishes and the relevant functions for the transition are η(ω) and Δ(η, ω). At this point we can start to talk about the effect of FA. When the non- resonant conditions dominate the non-homogeneous equations (1) and (8) are valid, but if left-hand material conditions emerge the equation that is valid is equation (3). Unfortunately, even when the solution of equations (1) and (8) can be expressed in terms of the resonant solutions, the step between the function η(ω) and the discrete spectrum η_e(ω) cannot be observed as a soft transition. Indeed, the condition for simultaneous solutions of the homogeneous and inhomogeneous equations is the orthogonality of the source term with the resonant solutions. But equation (8) can help us to understand the confinement of the electromagnetic waves mechanism.

In the last section we develop equation (8) that has the particular behavior to describe the transition between right-hand and left hand material conditions and converting itself in the appropriate kind of equation. The switching is generated by the FA, but then equation (8) must have the property to show how a confined field is liberated. Also in the past section we named precursors to resonances to the equation (4) solutions.

Because we are making the hypothesis that all the range of frequencies are taking into account, necessary a part of them include the so called evanescent waves, but then these are the precursors of the resonant waves. When the change of broadcasting conditions occurs, are precisely these waves whose are released and converted into travelling waves. So the structure of equation (8) shows how the process performance. First, necessary in equation (7) the third term vanishes that is,

secondly, for a continuity transformation of evanescent waves into travelling waves it is necessary that the equation (8) must have simultaneous solutions of the two kinds that is the source must be orthogonal to resonant solutions, and third not necessarily the two kinds of regimes are physically excluding (if the previous orthogonality condition is satisfied). All those conditions are accomplished because N. Von Der Heydt⁵ was shown that |$\mathop {\Delta (\eta (\omega))}\limits_{\lim \omega \to \omega _q } = 0$|$Δ(η(ω))limω→ωq=0$ and that |$\mathop {\eta (\omega)}\limits_{\lim \omega \to \omega _q } = 1$|$η(ω)limω→ωq=1$⁠, and we ask explicitly for the orthogonality.

Even when the formalism was developed thinking in their applications to the sending of information in a broadcasting process, we don't need to limit to an specific kind of system, for example, every device whose performance can be described as an input to output signals, where the evanescent electromagnetic waves and the resonances appear, can be analyzed with our view point. Very recently Xiang-ku Kong et al.⁸ studied the interaction of plasma with evanescent waves in a device whose behavior is like a left-hand material in which the effect is to increase the polarization by coupling the evanescent waves with the transmitted waves while by tuning the resonances with controlled magnetization we get left-hand material conditions. Because we cannot distinguish a point source in the region considered we neither can observe the orthogonality condition expressed later in equations (17) and (18), but we can observe de transition (coupling) from evanescent waves to travelling waves like an increase in the polarization effect. So we can also have the alternative of consider this experiment as an example of a media where we can switch the left-hand material conditions.

By the 4 × 4 matrices (not shown completely on this paper) developed by Xiang-ku Kong et al.,¹¹ we think that it could be possible, at least in principle, to obtain the kernel |${\sfb K}_n^{m( \circ)} (\omega ;\overline r,\overline {r}^\prime)$|$Knm(∘)(ω;r¯,r¯′)$ and build equation (1), and then it could be possible to build equation (3) and find the resonant frequencies by analytical procedure. Even if we won't follow this last suggested procedure on the plasma experiment, we can see in it, a real situation where evanescent waves and resonances play a fundamental role with the purpose of a better signal localization. Now, in Figure 1 we show a sketch of the Xiang-ku Kong et al., experiment:

In Figure 1 appear three plasma regions named U (unmagnetized), M (magnetized) and U (again unmagnetized). By changing the magnetization and the external voltage (not shown in the figure) it is possible to reach left-hand materials conditions. The relative permittivity tensor for U and M can be expressed as:¹¹ 

An example of the 4 × 4 matrix that describes an incident elliptically polarized beam that propagates along the z axis is:¹¹ 

For an explanation of the transfer matrix technique the authors Xiang-ku Kong et al., refers to Kato.^19,20

For normalization conditions (the auxiliary and original equations must have the same solutions) the structure of η(ω) can be selected as a phase factor, that is:

And when the resonant conditions are imposed, h(ω) must satisfy that

For the resonant frequency ω_q, where in general

So we must ask for h(ω_q) to be a real number.

The orthogonality condition for simultaneous solutions of both homogeneous and inhomogeneous equations is

But we have by using equation (3) that

And then the source must be related to the resonant solutions.

We can obtain an additional condition in case that the source was a point source, so equation (16) becomes

So we have

That is, the resonant solutions vanish at the point sources.

We have shown how the FA acts as a trigger of switching between positive refraction index-like conditions of broadcasting media to negative refraction index-like conditions. To this end we have constructed an auxiliary Fredholm's equation (equation (8)) that allows the knowledge of the transition process and suggests the structure of the Fredholm's eigenvalue. From general conditions we found that the eigenvalue function can be selected to have the structure of a phase factor (equation (10)). Also we have shown explicitly the orthogonality rule (equation (14)) for the transition between non-resonant to resonant conditions that tell us that the resonant solutions vanish at the point sources. This last condition then establish that if the source is an antenna, the resonant signal cannot come directly from this device, but we have an alternative when we define the so called information packs,^9,10,12 because for these packs the resonance are only convenient labels for information highways that guarantee the optimization of the broadcasting process. We briefly discussed the Xiang-kun Kong et al.¹¹ experiment in which we found a real support for three of the fundamental concepts of the present paper, that is, the switching of the left-hand material conditions (we establish that FA is the trigger of the changes), the presence of evanescent waves and the resonances. Although the last cited authors refer to a coupling between evanescent waves and resonances, we understand the same phenomenon as the transformation of evanescent (confined) waves in resonant transmitted (not confined) waves when we have left-hand material conditions. On appendix  A, we also proved that the integral equation we called a generalized Fredholm's equation with a generalized non-local source, is indeed a genuine Fredholm's equation and in appendix  B we prove that we can select the eigenvalue like a phase factor without lose their fundamental properties.

Because of the following statements: a) the transient process (between right-hand and left-hand materials conditions) depends only on the properties of the eigenvalue η(ω); b) the boundary conditions and the interaction information appear in the kernel |${\sfb K}_n^{m( \circ)} (\omega ;\overline r,\overline {r} ^\prime)$|$Knm(∘)(ω;r¯,r¯′)$⁠; c) the Fredholm's determinant can be written in terms of the singularities e as in equation (B12); and d) the conditions for the existence of resonances are equations (B10) and (B11). Then, as a corollary, we conclude that the eigenvalue η(ω), carry much information as |${\sfb K}_n^{m( \circ)} (\omega ;\overline r,\overline {r} ^\prime)$|$Knm(∘)(ω;r¯,r¯′)$⁠, although apparently doesn't have a spatial dependence. But it is obvious that we can write (B9) as

Equation (19) shows the relation between the eigenvalue η and the process of integration over the spatial variables in a manner which resembles the situation that occurs with an analytical function whose value in some interior point to a closed curve can be expressed with the aid of a Cauchy integral around this point and over the same curve.

Now we are going to show that the more generalized Fredholm's equation (8), with a generalized source can be viewed as the equation we obtained when we use the successive approximation method in a second step for the normal (but already in vector form) Fredholm's equation so we can recover the concept of a Fredholm's equation from extended definition the of a generalized Fredholm's equation.

If we try to solve equation (1) with |$f^m (\overline {r}^\prime,\omega) = f^{m( \circ)} (\overline {r}^\prime,\omega) + \varepsilon (\omega)g^m (\overline {r}^\prime,\omega)$|$fm(r¯′,ω)=fm(∘)(r¯′,ω)+ɛ(ω)gm(r¯′,ω)$ and |ε(ω)| ≪ 1, as a first approximation we have

And we can write for the second approximation

That can be written as

We remember that equation (1) is

Now we take the difference between equations (A3) and (A4):

So

But if we have a good approximation we can write

And this can be made if

So that equation (A3) can be written as

And then

Because we suppose that also |ε(ω)| ≪ 1 we can define

This function can be viewed as a small contribution because of the factor υ(ω)²ε(ω).

And then

The new source term has the same form of equation (7) with the factor υ(ω)²ε(ω) instead of Δ(η, ω)[η(ω) − υ(ω)] in the integral or non-local contribution. Indeed ε(ω) is expected to have a similar behavior to [η(ω) − υ(ω)]. So we have retrieved the concept of Fredholm's equation from the generalized Fredholm's equation with a generalized non-local source.

On this appendix we show how we can select the eigenvalue to have the structure of a phase factor, and to pursue this end, we recall some previous results.

From⁹ we have that

In this equation α_e(ω) is the Fredholm's eigenvalue that is the inverse of η_e(ω), and we can write their real and imaginary parts to have

Where

And we can define

So we can write

Now we can also define

We name the new functions |$V_e^m (\overline r ;\omega)$|$Vem(r¯;ω)$ renormalized functions.

It is clear that

Or that

Now, we know that N. Von Der Heydt⁵ demonstrated that the inverse of the Fredholm's eigenvalue satisfy

And that the conditions to claim solutions for equation (1) are:

Where

Then, conditions (B10) to (B12) allow us to obtain the renormalized solutions |$V_e^m (\overline r ;\omega)$|$Vem(r¯;ω)$ with the use of a phase shape eigenvalue and from this the original solution |$w_e^m (\overline r ;\omega)$|$wem(r¯;ω)$⁠.