IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 15, 2016 1681

Energy Accumulation in Waves Propagating inSpace- and Time-Varying Transmission Lines

Konstantin A. Lurie and Vadim V. Yakovlev, Senior Member, IEEE

Abstract—This letter introduces a new principle of energyaccumulation in transmission lines with the LC parameters vary-ing in space and time—an effect emerging from the concept ofdynamic materials. We consider a checkerboard material assem-bly created by a periodic temporal “flip over” of the propertiesof spatial laminates fabricated from two different substances. Insome extended ranges of material and structural parameters, anexponential growth of energy is maintained in waves travelingthrough this assembly due to the work done by an external sourceat the moments of temporal property switching. The effect isnumerically demonstrated for an equivalent LC circuit.

Index Terms—Capacitance, energy accumulation, inductance,spatial-temporal laminate, transmission line.

I. INTRODUCTION

F OR A long time, electromagnetic wave propagationthrough inhomogeneous media has been explored to

expand and advance microwave technology. Spatial nonunifor-mities were discussed in numerous publications, and temporalinhomogeneities became the subject of much fewer studies(e.g., [1]–[5]). In all of the above, the spatial and temporal vari-ations have been studied separately. Selected effects that arisefrom those variations combined have also been addressed bothin theoretical [6]–[8] and applied contexts [9]–[11]. A generalaccount of the phenomena emerging from space-time propertyvariability has been introduced as a special concept of the so-called dynamic materials (DM) [12]–[14]. However, a numberof important opportunities following from this concept withregard to microwave technology have not been given a detailedanalysis.

This letter outlines one of such opportunities: an originalapproach to electromagnetic energy generation and accumula-tion. DM are defined as substances with properties that maychange in space and time. They are not fabricated once forall, but rather brought into the scene and maintained by beingproperly operated both in space and in time. Specifically, thetemporal property change is possible due to energy exchangebetween the DM and the environment. A DM is a thermody-namically open system; its very notion makes sense only whenit is perceived as a spatial-temporal entity. Contrary to DM,ordinary materials appear to be thermodynamically closed.

Manuscript received October 20, 2015; revised December 04, 2015; acceptedJanuary 07, 2016. Date of publication February 08, 2016; date of currentversion September 02, 2016.

The authors are with the Department of Mathematical Sciences, WorcesterPolytechnic Institute, Worcester, MA 01609 USA (e-mail: klurie@wpi.edu;vadim@wpi.edu).

Color versions of one or more of the figures in this letter are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LAWP.2016.2522384

The DM concept could be materialized due to the propertytuning on both macro and micro scale. Some technical meansare detailed in [15] and [16], but the most simple and practicalway is a material switching, i.e., a conversion (replacement) ofone material into (for) another whenever and wherever neces-sary. The involvement of time in the transfer is accompanied bythe energy flux into the system or away from it; this exchange ismaintained by an external source. When one works with DM,the presence of such source is both imminent and critical: Itserves as a necessary link with the environment, preserving thedynamic nature of the material formation.

These ideas can be realized by building DM structured asarrays (transmission lines) of coupled and actively controlledoscillatory elements. In the electromagnetic context, one canthink of dielectric materials with tunable permittivity ε andpermeability μ; in a discrete version, the LC-cells may beconsidered instead. The DM represented as the tunable trans-mission lines offer a diversity of distinguishable features ofthe wave propagation through such structures. Mention shouldbe made, first and foremost, of their ability to dynamicallyamplify, tune, and compress signals over a wide range of fre-quencies. This letter is specifically aimed to demonstrate theeffect of energy accumulation and concentration produced byproper control of material properties in space and time.

II. THEORETICAL BACKGROUND

A. Continuous Transmission Line

Assume that we have two conventional isotropic nondis-persive dielectrics: material 1 and material 2, with the waveimpedance γ =

√μ/ε and phase velocity α = 1/

√εμ taking

values (γ1, α1) and (γ2, α2), respectively. Assume that γ1 = γ2and α2 > α1, so the wave impedances match, and the material2 (1) is fast (slow). Consider a periodic material laminate in 1-Dspace and time, with materials 1 (2) occupying the alternatinglayers with relevant volume fractions in every period. The lami-nate is called spatial (temporal) if the layers go perpendicularlyto the z-axis (t-axis) (Fig. 1). A spatial laminate is an ordinarymaterial assemblage in space, while a temporal laminate is aDM produced by a temporal property switching.

A plane electromagnetic wave with the wave vector in thez-direction exhibits no reflection on both spatial and tempo-ral interfaces [1], [4], [13]. When such waves travel througha spatial laminate, their energy remains constant in time. Whenthey travel through a temporal laminate, the energy increasesby the factor α2/α1 each time the wave enters fast material 2from slow material 1, and decreases by the factor α1/α2 when

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1682 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 15, 2016

Fig. 1. (a) Spatial δ-periodic and (b) temporal τ -periodic laminates.

Fig. 2. Spatial-temporal checkerboard structure; the insert demonstrates thesharpening of pulses.

it leaves material 2 and enters material 1. The net energy gainover a temporal period therefore is zero. The gain (loss) occursdue to the work produced by an external source and injectedinto the wave (or taken out of it). The source works in the pres-ence of the wave and therefore changes the material propertiesat each temporal property switching.

If we desire to accumulate energy in a traveling wave, lam-inates (Fig. 1) are not a solution. To secure accumulation,one has to change a material layout. The goal is to avoid themoments of energy loss by finding another wave route into theslow material 1. This route may be across the spatial, insteadof temporal, interface because when the wave crosses a spatialinterface, it does not lose energy since the energy flux remainscontinuous. We arrive at the key idea of testing a rectangularmaterial structure in space-time as a possible energy accumula-tor; the idea is crucial because it offers both spatial and temporalinterfaces and, as a consequence, a possibility to avoid the lossof energy.

Specifically, we try a “checkerboard” structure shown inFig. 2. This is a double periodic structure in (z, t)-plane, withperiods δ and τ , in z and t, respectively. To assemble it, webegin with a spatial periodic laminate, with period δ and volumefractions m and (1−m) of materials 2 and 1. This laminateexists from t = 0 to t = nτ (see Fig. 2), at which point itsmaterial properties “flip over”: Where there was material 1,there now appears material 2, and vice versa. This pattern ismaintained from t = nτ to t = τ , and the procedure repeatsperiodically in time, with period τ . It therefore appears that thisstructure may support the wave routes with no energy loss.

A remarkable property of the checkerboard is that it sup-ports the wave routes favorable for energy accumulation. Theycome into a series of groups converging each to its own selected

Fig. 3. Checkerboard structure in space-time with wave routes for α1 = 0.6,α2 = 1.1, m = 0.4, n = 0.5, and β = 1. Stable limit cycles and wave routesare shown as bold and thin broken lines, respectively.

wave route—a limit cycle [13]. Energy is focused in said groupsbecause they move from the fast material-2 cells into their slowmaterial-1 neighbors across spatial interface, and then proceedagain into the fast material across the temporal interface. Atthe first crossing, the energy remains continuous, while at thesecond, the energy is added into the wave. We thus avoid intru-sion into material 1 through the temporal interface at whichencounter the energy would be lost. The energy is concen-trated in high power pulses that sharpen as they move fromone double period to another (Fig. 2) (see [13] for the relevantmathematical exposition).

This peculiar performance of the wave routes is observedwithin some extended ranges of parameters m, n, α1, α2 [12],[14]. Adjusted to our notation, these ranges are specified by thefollowing set of inequalities:

α1 +(1− α1

α2

)mβ − β

α1 − α2≤ n ≤

α1 +(1− α2

α1

)mβ − β

α1 − α2

mβ − α2 +(1−m)α2β

α1

α1 − α2≤ n ≤ mβ − α2 +

(1−m)α1βα2

α1 − α2(1)

where β = δ/τ . These inequalities apply to homogeneousisotropic material constituents; they are sharp to the effect thatthe violation of one of them would destroy the concentration ofthe wave routes into groups converging to limit cycles.

The numerical experiment illustrated in Fig. 3 is relatedto propagation of plane electromagnetic wave with the wavevector along the positive z-axis through the checkerboardwith parameters m = 0.4, n = 0.6, α1 = 0.6, α2 = 1.1, β = 1.These values fall into the ranges (1); specifically, inequalities(1) imply: 0.44 < 0.6 < 2.57, −0.8 < 0.6 < 0.75. There mayalso be another wave traveling in the negative z-direction andtotally independent of the previous wave because of the alter-nating layers with relevant volume fractions in every period.

As seen from Fig. 3, after a few introductory periods, thewaves enter the fast material 2 across the temporal interfacesat which they gain energy, and leave it (i.e., enter material 1)across the spatial interface where no energy loss occurs. Energyis accumulated in narrow pulses that carry high power. Thecycles are stable because they attract the neighboring wave

LURIE AND YAKOVLEV: ENERGY ACCUMULATION IN WAVES PROPAGATING IN SPACE- AND TIME-VARYING TRANSMISSION LINES 1683

Fig. 4. Sketch of discrete version of spatial-temporal checkerboard structureimitating one double period of the structure in Figs. 2 and 3.

routes. Two consecutive stable limit cycles are separated byan unstable limit cycle that repels its neighboring wave routes;such cycles are not shown in Fig. 3. For one period, the energyof the wave increases by factor (α2/α1)

2 because it is injectedtwice into the wave; as a result, the energy of the wave growsexponentially in time.

This mechanism of generation is different from the para-metric resonance practiced in the traveling wave parametricamplifiers [9], [10], where a “pump” wave is propagated tochange the shunt capacities in the transmission line. Thischange occurs to just one material property in a spatial-temporal laminate; the system remains resonant and requiresfrequency adjustment to be able to supply energy for ampli-fication. Unlike that, the checkerboard structure is free fromreflections and therefore nonresonant and requiring no fre-quency adjustment. The energy accumulation occurs for allfrequencies provided that the material and structural parame-ters belong to the domain (1) of generation. This guaranteesrobustness of the checkerboard assembly. The absence of reflec-tion is due to the involvement of both dielectric and magneticparameters that secure matching of the wave impedance ofparticipating materials.

B. Discrete Transmission Line

A continuous checkerboard structure may be implementedas a standard transmission line with linear inductance L andcapacitance C taking values (L1, C1), (L2, C2) related to twodifferent materials. The equations of a transmission line

∂V

∂z= −∂(LI)

∂t,∂I

∂z= −∂(CV )

∂t(2)

introduce the voltage V and current I instead of the electricand magnetic fields in the equations of a plane electromagneticwave, and L and C as replacements for μ and ε. On spatialproperty interfaces, we require continuity of V and I , whileon temporal interfaces, the magnetic flux LI and charge CVshould be continuous.

One feasible discrete representation of a continuous trans-mission line is a spatial periodic array of sections (groups)assembled each from many LC-cells, with properties L1, C1

and L2, C2 alternating from section to section (Fig. 4). A spa-tial period is represented by two groups of cells: sections withproperties L1, C1 (L2, C2) imitate material 1 (2). Cells andtheir groups are connected in series; they are tunable, with the

Fig. 5. Flowchart of a simplified version of transmission line (Fig. 4) used incomputation (Fig. 6).

choice of properties in each determined by switch S. By oper-ating the switch, material 1 becomes material 2, and vice versa.Particularly, if the switches are operated periodically in time,then the construction acquires a temporal period and can betreated as a checkerboard in space-time. The difference in anumber of cells in the groups (Fig. 4) imitating materials 1 and2 (Fig. 2) reflects the difference in the volume fractions (m,1−m) in a spatial period at different parts (n, 1− n) of thetemporal period represented by two rows of cells in Fig. 4.

III. ILLUSTRATIVE RESULTS

Numerical results presented below are related to a simpli-fied version of the circuit in Fig. 4. This version assumes thatthere are only two consecutive arrays exhibiting temporal prop-erty change at proper times (Fig. 5). The arrays are connectedat point B that plays the role of spatial interface. Assumethat over the fraction n of the temporal period τ , the arraysI and II imitate, respectively, materials 2 and 1, with relatedswitches (Fig. 4) held in corresponding positions. Through therest (1− n)th fraction of τ , the switches are kept in oppositepositions, so the material properties “flip over” from 1 to 2,and vice versa, in both arrays. Constructively, two arrays workmore than once in a single run. In the time interval (0, nτ ),the pulse enters the array I occupied, at that time, by material2, through the spatial input I (Fig. 5). Then, it proceeds intoarray II, i.e., into material 1, again, through a spatial interfaceB. At some point during this run, the input I disconnects fromthe pulse source and connects to the output O of array II (ter-minals I1 and O1, as well as I2 and O2, coming into pairwisecontact). The network becomes a loop—the spatial transitionsoccur from one material to another.

We require that the entire pulse has to “fit” in array II at theend of the previous run, so that the temporal switching frommaterial 1 in array II to material 2 in the same array can occurat that exact moment. However, the number of LC-cells canbe cut in half (with the same result) if the pulse is assumedsymmetric, and we use its reflection from the shorting (or float-ing) end of the second array. The reflected wave has oppositepolarity and travels in the opposite direction. At the momentwhen two contrary waves come in opposite phase, the tempo-ral switching occurs, and the energy stored in inductances (orcapacitors) is pumped into the wave now traveling in the oppo-site direction. The underlying theory allows to work with eithertype of energy [13]: The power accumulation still develops, andthe wave impedance is preserved as long as temporal switchingin both of L and C preserves the ratio of these parameters.

In the illustrative computation performed with the use ofOrCAD EE PSpice, we consider 32 LC-pairs in each array;the linear L and C are defined as (45 μH/m, 450 pF/m)for the fast material, and (55 μH/m, 550 pF/m) as similar

1684 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 15, 2016

Fig. 6. (a) Pulse propagation through a “loop” combined from two mediaalternating in space and (b) evolution of pulses in the presence of temporalswitchings.

properties for slow material. The phase velocities have the val-ues α1 = 1/

√LC = 5.75 · 106 m/s and α2 = 7.03 · 106 m/s,

respectively; the ratio α2/α1 = 1.22. The common value ofthe wave impedance γ =

√L/C equals 316 Ω. Fig. 6(a) illus-

trates the case of wave propagation through a spatial laminate(no temporal property change). The curve shows the waveformobserved at the point B (Fig. 5) between the two materials. Thepositive pulses propagate from left to right, and the negativeones travel from right to left. It is seen that after nine passesthrough the arrays, the pulse retains its shape reasonably well.

In Fig. 6(b), there are eight temporal switchings, so thereis total energy accumulation by factor of 1.228 ≈ 4.9. Thetotal energy trace (shown by dashed curve) gives at the endof the graph the value of ∼ 0.94 J. The energy of the initialpulse is specified as 0.19 J. That gives a total accumulation of0.94/0.19 ≈ 4.95, which is very close to the expected value of4.9. Inequalities (1) apply to this case, with β = δ/τ chosen as(α1 + α2)/2 = 6.39 · 106 m/s; as a result, they take the form

0.5− 0.91m < n < 0.5 + 1.11m

−0.6 + 1.11m ≤ n ≤ 1.45− 0.91m. (3)

The choice m = 0.4, n = 0.6 (the same as in Section II-A) matches these inequalities. The ratio β = 0.064 · 108 m/simplies the spatial period 0.128 m for 50 MHz switching fre-quency [Fig. 6(b)]; alternatively, this period becomes 0.64 mfor 10 MHz switching frequency.

The discrete version of a transmission line tuned to fit theabove theory demonstrates the phenomenon of energy accumu-lation in sharpening pulses. At the same time, this version intro-duces some high-frequency oscillations visible in Fig. 6(b) and

created by individual LC-cells. These frequencies disappear ina continuous model that is nonresonant in the absence of reflec-tions. In a discrete version, there arises a tradeoff between thematerial scale of LC-cells and the energy accumulation; it canbe compromised by appropriate engineering design.

IV. CONCLUSION

A novel principle of energy accumulation in the pulses gen-erated in material formations with space- and time-varyingparameters has been introduced. This effect arises in a checker-board material structure in 1-D space and time. Discrete versionof such structure has been suggested as a transmission line com-posed of the groups of consecutive LC cells having propertiesthat alternate from group to group. A numerical experimentconfirms the main theoretical result: The energy is accumulatedand demonstrates exponential growth in the series of travel-ing and sharpening pulses. Future theoretical work will clarifyhow the negative effects produced by such factors as inevitablelosses and/or not ideal match of the wave impedances may bereduced by proper control of material and structural parame-ters of the checkerboard. Future experimental work will identifyapplications and devices in which the proposed approach maybe particularly beneficial.

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