An impedance approach to the response of matter

At the dawn of the research on waveguides the propagation by electrical conduction through transmission lines was compared with the general transmission theory of plane electromagnetic waves. After the invention of lasers we wonder whether the impedance concept can allow a seamless shift from the geometric rendering of received electromagnetic signals to the understanding of simple arguments on power transfer. Perhaps, the impedance concept could help noticing the occurrence of radiation-matter interactions and give hints as to how some phenomena could be enhanced by exposing matter to specific non-ionizing radiations.


Introduction
Over time radiation has been tted with a number of attributes, among which energy stands out. Current mathematical theories, primarily electromagnetism, the theory of relativity, and QED, are mainly concerned with two types of tasks. Firstly, the representation of the distribution and propagation of the electromagnetic (EM) eld in physical space, which is faced by drawing on geometry, and secondly, the investigation of the interaction of radiation with matter. In a matterdominated universe natural radiations are perceived as only slightly aecting matter, perhaps except for ionizing radiations. This attitude transpires from the analysis of quantum eects. It doesn't imply that solar radiation elicits small eects on matter, but rather that those don't pop up stamped with the ω radiation-matter interaction seal. While up until WWII nobody felt the need to extend the theoretical framework beyond the perturbative approximation, researchers gradually started pointing out nonlinear interaction eects after the technical achievements in the generation, manipulation, and detection of radiation. In the end all types of phenomena appear to show a degree of individual variability. Thus, small eects are being isolated, that were previously overlooked or unnoticed, and whose explanation demands some kind of nonlinear approach. Emergence usually needs either a more detailed case-by-case description, or supplementary ad hoc assumptions, and attempts are ongoing at explanations based on statistical inference and the extant probabilistic theories. However, more often than not alongside`nonlinearity' radiationmatter interactions feature specicity. Alternatively, one can deem variability to be worth a second look when availing of directional sources, power ampliers, sensitive receivers, and computer assisted data collection and analysis. Perhaps, phenomena that on Earth take place under the inuence of sunlight manifest dierently in other environments, and can be enhanced under articial conditions. Illumination is relevant to our life, and understanding how it may aect matter can open new avenues. In the following we'll discuss a few points related to an impedance approach to the response of matter.

Electromagnetic (EM) eld as a property of space
Human eyes receive in the so-called visible range (2.5 − 5) · 10 15 rad s . Although the luminous eciency is not uniform in this range, for luminance levels in the 10 − 10 7 lux range the adapted photopic response of the normal eye can be described by a linear function [1]. In this range radiations are ineable as they reveal bodies without being themselves visible. Furthermore, the eyes are not for direct viewing of light sources. Rather, most seen images consist of passively scattered radiations by lit bodies pertaining to the outer world, that happen to be received by the sense organs. Although our eyeballs function like EM signal receivers, our visual perception is far from linear. Humans tend to focus attention on the details of interest with the purpose of recognizing them and interpreting information, decidedly brain activities. Notwithstanding this, while standing still the eye movements allow sensing a`whole space' around the particular that initially caught the eye. And, intriguingly enough, we perceive the visualized feature as located in space. The ancients formulated geometry in order to deal with shapes in space. Owing to Euclid's involvement in optics we know that to him mirrors likewise focus on details. In contrast, when diaphragms and vignetting can be disregarded our optical image forming instruments, above all lenses and mirrors, limit to render what we call`space'. Sizes and aspects of all items of surveyed space change considerably when recorded from dierent locations, or when dierently lit. Images also keep changing under minor optical alignment adjustments. This is shown in Figure 1. On closer attention, those changes appear to be due to the inherent inability of optical instruments (microscopes, radars, receiver antennas, and whatnot) to select and follow the detail of interest.
Provided we clarify what mathematical framework we mean by`received space', and achieve a reasonable rendering of it, visual geometric representations can enable a coherent interpretation of the perceived objects across the whole domain of instrumental imaging. This, in turn, can ease the detection of nonlinear traits. The EM eld that came to be understood as the locus of EM phenomena evolved from Faraday's magnetic lines of force. Faraday recorded the action of magnetic forces on test bodies (TBs) at dierent locations disregarding the functioning of the detectors, namely, the position of iron lings on a paper sheet or the impulse response of a coiled conductor wire (Faraday induction). Experimentally, the EM response depends on the source of the eld, e.g. the magnet, and on the receiver impedance Z t . At worst, it includes a non-quantiable contribution of the interposed medium. When mapping the spatial distribution of the`stu' constituting the magnetic eld, we are the terminal eld sensor locating the TB in space in daylight, and thus Z T B = Z t . Experimental data points can be analyzed in terms of the small-signal equivalent electric circuit (EEC) of the TB, ling or coil, characterized by a time-invariant EM impedance 3 Z T B . Let's resort to a TB consisting of a RLC series circuit, and draw on hydrodynamic analogies. The impedance Z T B = V I , a pressure/ux analog, belongs to the EEC diagram of the TB whose voltage and current are respectively V and I. Moreover, let's neglect that Faraday observed free induction decays and dwell instead on the oscillatory behavior of the RLC connected to an electromotive force (EMF). In the real domain R the time response of the EEC is obtained by applying Kirchho's voltage law V R (t) + V L (t) + V C (t) = E(t), i.e. by solving the complete second order ordinary dierential equation (ODE) with real constant coecients R(resistance), L(inductance), and C(capacity) that models the TB. The steady state impedance Z T B is a complex function even when the circuit is driven by the EMF . Since integrals over odd terms vanish, if an inverse Fourier transform h(t) = 1 2π ∞ −∞ H(ω)e iωt dω existed in this case, it would be a real function. The energy interpretation of the EM eld analyzes the collected TB data in terms of EEC by drawing on a hydrodynamic analogy. Qualifying the TB data as properties of space takes advantage of our spatial perception.

Resonance, a nonlinear coupling condition
What do we perceive, or measure with an image forming device, when a TB tightly couples to a radiation source, i.e. when the source is tuned to it? The principle of maximum power transfer tells that, for source and TB to couple, the allegedly time-invariant Z T B of the two-port EEC of the TB ought to match the source. Writing R for the voltage reection coecient, and n for the refractive index, the TB's reectance is |R| 2 = Z T B −Zsource Z T B +Zsource 2 = nsource −ns nsource +ns 2 = 0 under matching conditions. Ideally, its interface and internal structure may be deeply aected without scattering light, and we see nothing. For an example, at very high frequencies (ionizing radiations) n → 1 for all matter, and matched conditions possibly occur. Similar conditions apply to the sun as a light source, and cause seen objects to look like`mismatched' loads. Still, the source itself cannot be modeled as an EEC, being it an active element. Moreover, we expect the TB to respond nonlinearly, too, when a matching condition is steadily approached, and specically to start ickering. 3 In fact, engineers often choose the source impedance of a driving stage Z t much smaller than the load just to avoid aecting the voltage transferred to the receiver stage. The linear signal approach is awless as long as icker contributes a minor EM response, and the nal receiver doesn't itself couple. Yet, resonance is a nonlinear phenomenon. Although a scant power may suce for ecient coupling, to uphold the matched condition an articial source ought to follow the time-uctuations in Z T B .

Lasing sources and geometry
Poynting likely addressed the problem of the geometric representation of the EM energy transfer in the context of the`null system' [2]. 4 Figures 1C-D show that no directional light readily lends itself to a geometric interpretation. Rather, the attendant things need to diuse to reveal the outer world. Lasers are reckoned to be directional sources on a par with direct sunlight. Here we won't digress to consider how the sensed shapes depend on illumination, and just stress that the transfer eciency of sources depends on impedance matching. EM sourceobject coupling famously occurs through an interposed void, as opposed to what happens with mechanical analogies. This is why individual ne-tuning can fool statistical predictions and energy interpretations. In principle, an impedance approach is suited to discuss diuse and directional illumination. To that end, let's briey retrace the cross-fertilization of engineering and theoretical developments that makes innite transmission line and eld equations look the same [3]. An ideal transmission line is ruled by equations dV dz = −ZI and dI dz = −YV . Its electric parameters R, L, G, C allow writing series impedance Z = R + iωL, shunt admittance 5 besides other related functions, such as the reection and transmission coecients. As with the two-port RLC network and in optics it is possible to pair output and input functions via Fourier/Laplace transform. Other types of transforms have been developed in the top-down network synthesis of RLClters. To discuss the propagation along the line in terms of a linearly polarized TEM wavefront parametrized by the elds E =xE 0 e −γz and H =ŷ E 0 η e −γz Schelkuno replaces the line equations by dE x dz = −iωµH y and dH y dz = −(g + iωε)E x , respectively. The propagation constant γ and the intrinsic impedance η are written out in terms of the wave parameters permittivity ε, conductivity g (ideally, g = 0), and permeability µ. They become γ = iωµ · (g + iωε), Z 0 → η = iωµ g+iωε = γ g+iωε = iωµ γ . Once this is done, Z 0 neither transforms as a vector, nor satises the addition rule. This problem has been tackled by various authors [4]. However, if Z is related to streamlines and equiphase surfaces, it allows for transformations over C in`space' as an entire analytic function [5]. Starting with the so-called stereographic projection, 6 the representation of C -numbers as points can be interchangeably represented either on the Argand plane, that as a eld can be made isomorphic to R 2 , or on the Riemann sphere S 2 (in R 3 ). 7 So can level curves of entire analytic functions over C. Signal and power transfer can both be dealt with in terms of Z surfaces/spots without reifying the elds. In given experimental conditions the energy issues can be rated with respect to conversion yields.