ACCEPTED AND TO APPEAR IN THE IEEE MICROWAVE …

ACCEPTED AND TO APPEAR IN THE IEEE MICROWAVE MAGAZINE 1

Space-Time Modulation:Principles and Applications

Sajjad Taravati and Ahmed A. Kishk, Fellow, IEEE

I. INTRODUCTION

The ever-increasing demand for high data rate wirelesssystems has led to a crowded electromagnetic spectrum. Thisdemand has successively spurred the development of versa-tile microwave and millimeter-wave integrated componentspossessing high selectivity, multi-functionalities and enhancedefficiency. These components require a class of nonreciprocalstructures endowed with extra functionalities, e.g., frequencygeneration, wave amplification and full-duplex communica-tion. Space-time (ST) modulation has been shown to be aperfect candidate for high data transmission given its extraordi-nary capability for electromagnetic wave engineering. Space-time-modulated (STM) media are dynamic and directionalelectromagnetic structures whose constitutive parameters varyin both space and time. Recently, STM media have receivedsubstantial attention in the scientific and engineering com-munities. The unique and exotic properties of STM mediahave led to the development of original physics concepts, andnovel devices in acoustics, microwave, terahertz and opticareas. STM media were initially studied in the context oftraveling wave parametric amplifiers in the 50s and 60s [1]–[6]. In that era, magnet-based nonreciprocity was the dominantapproach to the realization of isolators, circulators and othernonreciprocal systems. However, magnet-based nonreciprocityis plagued with bulkiness, nonintegrability, heaviness andincompatibility with high frequency techniques.

Recently, ST modulation has experienced a surge in scien-tific attention thanks to its extraordinary and unique nonre-ciprocity. It eliminates issues of conventional nonreciprocitytechniques, such as bulkiness, heaviness and incompatibilitywith integrated circuit technology associated with magnet-based nonreciprocity, power restrictions of nonlinear-basednonreciprocity, and frequency limitations and low power han-dling of transistor-based nonreciprocity. It provides asymmet-ric interband photonic transitions [7]–[14], subluminal andsuperluminal phase velocities, and asymmetric dispersion di-agrams [4], [11], [13], [14], and holds potential for energyaccumulation [15]. Various enhanced-efficiency magnet-freemicrowave and optical components have been recently realizedby taking advantage of the unique properties of ST modulation,including isolators [9], [11], [16]–[19], circulators [20]–[23],

Manuscript accepted for publication in the IEEE Microwave MagazineOctober 19, 2018.

Sajjad Taravati ([email protected]) is with the Edward S. RogersSr. Department of Electrical and Computer Engineering, University ofToronto, Toronto, Ontario M5S 3H7, Canada.

Ahmed A. Kishk is with the Department of Electrical and ComputerEngineering, Concordia University, Montreal, QC H3G 2W1, Canada. (e-mail:[email protected])

pure frequency mixer [24] metasurfaces [25]–[28], one-waybeam splitters [12], [14], nonreciprocal antennas [29]–[31],and advanced wave engineering [32].

This paper overviews the principles, theoretical analysis andnumerical simulation of STM media and their applications incommunication systems. Fig. 1 presents an overview of theorganization of the paper. We first present a global perspectiveon the principles of ST modulation, naturally and artificiallycreated STM media and their applications in communicationsystems. Next, the interaction of artificial STM media withelectromagnetic waves is analyzed, and numerical simulationof electromagnetic wave propagation and scattering from thesemedia is provided. Next, the dispersion diagram of the STMmedium is studied for the characterization of subluminal,luminal and superluminal ST modulations. In addition, weinvestigate the effect of length, modulation velocity, modu-lation frequency, and equilibrium on wave scattering fromthe STM medium. The extraordinary interaction of the STMmedium with electromagnetic waves leads to unidirectionalfrequency generation and amplification, which are leveragedfor the realization of various enhanced-efficiency broadbandmicrowave components that are integrable with circuit tech-nology, lightweight, tunable, and low cost. These applicationsinclude, but are not limited to, magnet-free isolators, magnet-free circulators, transceiver front-ends, and pure frequencymixers.

II. PRINCIPLES OF ST MODULATION

A. Natural Phenomena Due to ST Modulation

Space and time represent general forms of the existenceof matter. Space possesses three dimensions:length, widthand height, whereas time is represented by one dimension–from the past through the present to the future–which isunrepeatable and irreversible. ST modulation leads to variousnatural phenomena. For instance, consider Fig. 2. The waterwaves in the ocean are unidirectional and possess very longwavelengths with small amplitude. However, flying an airplaneon top of the ocean along the same direction as the waterwaves leads to ST modulation of the water. This, in turn,yields tremendous amplification of the water waves. The samephenomenon occurs in the ocean when a strong wind blowsalong the same path as the water waves and leads to hugewater waves at the ocean’s surface.

A tsunami is another natural phenomenon caused by STmodulation of water waves (Fig. 3). Tsunamis, or seismic seawaves, are giant waves caused by earthquakes, specificallysubduction zone earthquakes, occurring under the ocean. The

arX

iv:1

903.

0127

2v2

[ph

ysic

s.cl

ass-

ph]

12

Mar

201

9


Pure frequency mixer

Natural phenomena

due to ST modulation

Principles of ST modulation

(Sec. II)Sinusoidally STM media

(Sec. III)

Applications of ST modulation

(Sec. IV)

Analysis of general

periodic STM media

General sinusoidally ST

ε- and/or µ-modulated

medium

Subluminal, luminal

and superluminal

ST modulations

Numerical simulation

and parametric study

Nonreciprocal scattering

Superior performance of

equilibrated ST modulation

Magnet-free linear isolators

Quasisonic isolator

Asymmetric photonic

band gaps isoator

Self-biased broadband

Isolator based on

unidirectional ST coherency

Magnet-free circulator

Transciever front-ends

Broad-band low-loss

nonreciprocal component

Nonreciprocal transmitter

(one-way mixer-antenna)

Full-duplex transciever

(mixer-duplexer-antenna)

Fig. 1. Organization of the paper.

Direction of airplane

(ST modulation)

Amplified water waves

Direction of water waves

Fig. 2. An airplane flying on top of the ocean, spatiotemporally modulates the water waves, resulting in tremendous amplification of the waves [33].


movement of the tectonic plates due to the earthquake alsoresults in the movement of the sea water. A major earthquakeunder the ocean’s floor creates shock waves which push thewater up, leading to ST modulation of the water. As a conse-quence, at the shore, the height of the water waves increasesenormously as the speed decreases, causing the tsunami. Thevelocity of a tsunami depends on the depth of the ocean overwhich it is traveling, as well as the depth of the earthquake.If the earthquake occurs more than 100 km below the oceanfloor, a tsunami will not occur because of insufficient verticaldisplacement of the water.

B. Analysis of General Periodic STM Media

In physics and engineering, ST modulation is created byspatiotemporal modulation of the constitutive parameters ofmedia, e.g., electric permittivity [4], [9], [11], [34], magneticpermeability [2], [13], [35], or electrical conductivity [22],[23], [36]. Such ST modulation has been realized by a time-varying modulation signal that propagates along the mediumand spatiotemporally modulates variable capacitances/induc-tances which are added to the intrinsic capacitance/inductanceof the background medium [11], [34]. A general STM slab isdepicted in Fig. 4 [11], [13]. Such a general ST medium isrepresented by a unidirectional refractive-index

n(z, t) = f(βmz − ωmt), (1)

where f(.) is arbitrary periodic functions of the ST phasevariable ξ = βmz − ωmt, and βm and ωm are respectivelythe spatial and temporal modulation frequencies. Since therefractive-index of the STM slab is unidirectional, we in-vestigate the electromagnetic wave transmission for the waveincidence from the left and right sides of the STM slab sepa-rately, called the forward (“F”) and backward (“B”) problems,throughout this paper. The forward problem is characterizedby an incident wave impinging on the slab from the left sideEF

I = xA0e−i(k0z−ω0t), and the transmitted wave from the

opposite side of the slab, EFT; while the backward problem is

represented by the incident wave impinging on the slab fromthe right side EB

I = xA0ei(k0z+ω0t), and the transmitted wave

from the opposite side of the slab EBT .

The spatial modulation frequency βm is related to thetemporal modulation frequency ωm, as

βm =ωm

vm=ωm

γvb, (2)

where vm is the phase velocity of the ST modulation whichmay be smaller or greater than the phase velocity of thebackground medium vb = c/

√εrµr, with c = 1/

√µ0ε0 being

the speed of light in vacuum, and with εr and µr denoting therelative permittivity and permeability common to regions 1and 3. In Eq. (2), γ is the ratio between the modulation andbackground phase velocities which is called the ST velocityratio. The limit γ = 0 corresponds to a purely space-modulated medium, whereas the limit γ =∞ yields a purelytime-modulated medium [11], [37], and γ = 1 corresponds tothe STM medium where the modulation propagates exactly atthe same velocity as a wave in the background medium.

The general periodic STM refractive-index n(z, t) in (2)may be represented in terms of general periodic STM electricpermittivity and magnetic permeability, i.e., ε(z, t) and µ(z, t).Due to the spatial and temporal periodicity of the permittivityand permeability, they may be expressed in terms of STFourier series expansion, i.e.,

ε(z, t) =

∞∑k=−∞

εke−jk(βmz−ωmt), (3a)

µ(z, t) =

∞∑k=−∞

µke−jk(βmz−ωmt), (3b)

where εk and µk are the coefficients of the kth term, and ε0 andµ0 are the average permittivity and permeability of the STMslab. Assuming TMyz or Ex polarization, the electromagneticfields inside the slab, i.e., ES(z, t) and HS(z, t), may berepresented in the double ST Bloch-Floquet form

ES(z, t) = xe−i(β0z−ω0t)∞∑

n=−∞Ene

−in(βmz−ωmt), (4a)

HS(z, t) = ye−i(β0z−ω0t)∞∑

n=−∞Hne

−in(βmz−ωmt), (4b)

where β0 and ω0 represent, respectively, the spatial andtemporal frequencies of the fundamental harmonic, n = 0.

In Eq. (4), the given parameters are the amplitude of theincident field (fundamental harmonic) A0, the spatial andtemporal frequencies of the incident field (k0 and ω0), thespatial and temporal modulation frequencies (βm and ωm), theperiodic function f(.), and length of the slab L. We achieve thedispersion relation β0(ω0) and the electromagnetic field solu-tions inside (En and Hn) the STM slab in Fig. 4 by satisfyingMaxwell’s equations. Since the slab assumes no variation inthe x- and y-directions, ∂ES/∂x = 0 and ∂ES/∂y = 0. Hence,∇ × ES = y∂Ex(z, t)/∂z and ∇ × HS = −x∂Hy(z, t)/∂z.As a result, the sourceless Maxwell equations read

∂Ex(z, t)

∂z= −∂ [µ(z, t)Hy(z, t)]

∂t, (5a)

∂Hy(z, t)

∂z= −∂ [ε(z, t)Ex(z, t)]

∂t. (5b)

The scattered electromagnetic fields, i.e., the reflected fieldsEF,B

R and HF,BR and the transmitted fields EF,B

T and HF,BT

are achieved by applying the boundary conditions for thecontinuity of electromagnetic fields at z = 0 and z = L [11],[13]. Fig. 5 presents the procedure for achieving unknown fieldamplitudes and dispersion relations.

III. SINUSOIDALLY STM MEDIA

A. General Sinusoidally ST Permittivity- and/or Permeability-Modulated Medium

Sinusoidal ST modulation is the most common and appli-cable form of ST modulation. Such a sinusoidal and unidirec-tional ST modulation is achieved using variable capacitancesand variable inductances, distributed at a subwavelength levelon a transmission line, which are spatiotemporally modulated


Fig. 3. A tsunami as a natural phenomenon resulting from ST modulation of water waves via an underwater earthquake, so that the water waves at the shoreacquire short wavelength but strong amplitude.

Fig. 4. Electromagnetic scattering from a periodically STM slab (region 2) sandwiched between two semi-infinite unmodulated media (regions 1 and 3). Dueto the unidirectionality of the modulation, ε(z, t) = fper(βmz − ωmt), the system is nonreciprocal, with different temporal and spatial frequencies scatteredin the two directions [11], [13].

by a +z-propagating harmonic wave generator with temporalfrequency ωm [2], [11], [13], [34], [35]. Practical realiza-tion of variable inductors (permeability) has been extensivelydiscussed in [13]. It has been shown that voltage-controlledvariable inductors may be realized using varactors and thyris-tors [38]–[42]. In addition, spatiotemporal modulation of thepermeability has been experimentally demonstrated in [35].Here, we consider the general case of a sinusoidal STMtransmission line (Fig. 4), represented by the STM permittivityand permeability

ε(z, t) = εav [1 + δε cos(βmz − ωmt)] , (6a)

µ(z, t) = µav [1 + δµ cos(βmz − ωmt)] , (6b)

interfaced with two semi-infinite unmodulated regions. In (6),εav and µav respectively denote the average electric permittivityand average magnetic permeability of the transmission line,and δε and δµ respectively represent the amplitudes of the

permittivity and permeability ST modulations. The sinusoidalST modulation in (6) may be decomposed into its ST Fouriercomponents in (3), i.e., ε−1 = ε+1 = εavδε/2, µ−1 = µ+1 =µavδµ/2, ε0 = εav and µ0 = µav.

B. Subluminal, Luminal and Superluminal ST Modulations

The wave propagation phenomenology within the STM slabmay also be studied by virtue of the dispersion diagrams ofthe corresponding unbounded medium. The dispersion relationis represented by β0(ω0) which is achieved following theprocedure given in Fig. 5, that is, by injecting the Bloch-Floquet decomposed electromagnetic fields into Maxwell’sequations and taking the corresponding space and time deriva-tives [4], [13], [34]. In the limiting case of a vanishingly smallmodulation amplitude, δε = δµ → 0, the dispersion relation forthe sinusoidal ST modulation will be simplified to the closed-


Modulation parameters,

β and ω m m

Incident field parameters,

A , k and ω 0 00

Form the Bloch-Floquet

representation of

EM fields in 4(a) and 4(b),

E (z,t) and H (z,t)

Form the Fourier series

representation of

ε and µ in 3(a) and 3(b),

ε(z,t) and µ(z,t)

Modulation function,

f(.) for ε and µ

Maxwell’s equations

in 5(a) and 5(b)

S S

Achieve the dispersion

relation β (ω )

0 0

Application of

boundary conditions for E and H

at z=0 and z=L, separately for

forward and backward problems

Length of the slab,

L

Achieve the reflected and transmitted

EM fields (E , H , E and H ) and

amplitude of the fundamental harmonic

inside the slab,

E and H

Achieve the normalized unknown

field coefficients in 4(a) and 4(b),

E /E and H /H n n0 0

0 0

R R T T

Fig. 5. Procedure for achieving the scattered electromagnetic (EM) fieldsolutions in the STM slab in Fig. 4.

form dispersion relation [11], [13]

β±0βm

= γω0

ωm+ n (γ ∓ 1) . (7)

where β+0 and β−0 represent the spatial frequency of the fun-

damental space-time harmonic (STH) (n = 0) for forward andbackward, respectively. The ST modulation may be classifiedin three different categories, i.e., subluminal (vm < vb or0 < γ < 1), luminal (vm = vb or γ = 1) and superluminal(vm > vb or γ > 1) ST modulations, where vb = cif the background medium is air. Figs. 6(a), 6(b) and 6(c)plot the analytical dispersion diagrams of the STM mediumof the aforementioned three ST modulations. The dispersiondiagram of periodic STM media is constituted from an infiniteperiodic set of β±0 /βm−ω0/ωm straight lines, labeled n [11],[13]. Each curve represents a mode excited at the incidentfrequency ω0, as well as the oblique STH of another mode,excited at another frequency. For a homogeneous non-periodicunmodulated medium with ωm = βm = 0, n = 0 is theonly remaining curve, and for a vanishingly small pumpingstrength, δε = δµ → 0, the medium is quasi homogeneous,where most of the energy resides in the fundamental forwardand backward STH n = 0.

Fig. 6(a) plots the dispersion diagram of the subluminal(space-like) ST modulation, with δε = 0.4, δµ = 0 andγ = 0.5. It may be seen from this figure that the forwardand backward STHs acquire different horizontal distances, i.e.,∆β± = β±n+1 − β±n so that increasing γ yields an increase in

the ∆β− and a decrease in the ∆β+. For a static medium,vm = 0 and ∆β+ = ∆β−. However, as vm increases, theforward and backward harmonics experience different veloc-ities relative to the modulation wave, i.e. v+ = vb − vm andv− = vb + vm, respectively [11], [13]. As a consequence, theST transitions (exchange of energy and momentum betweenSTHs) for the forward STHs are stronger than those of thebackward STHs, revealing a nonreciprocal wave transmissionfor forward and backward wave incidences. It may be observedfrom Fig. 6(a) that for a nonzero modulation strength, i.e.,δε 6= δµ ≥ 0, the subluminal ST modulation provides verticalω-band-gaps at the synchronization points between forwardand backward harmonics.

Fig. 6(b) plots the dispersion diagram except for the luminalST modulation, i.e., γ → 1 and δε = δµ → 0. In thiscase, the forward harmonics acquire the minimum distanceof ∆β+ → 0, while the backward ST harmonics acquire themaximal distance of ∆β− → 2. Hence, a strong nonreciprocaltransmission for forward and backward incident fields isprovided by the STM medium. For the superluminal (time-like) ST modulation in Fig. 6(c), similar to the subluminalST modulation in Fig. 6(a), the forward and backward STHsacquire different horizontal distances ∆β± = β±n+1− β±n , butincreasing γ leads to a decrease in the ∆β− and an increasein the ∆β+. In contrast to the subluminal ST modulation,the superluminal ST modulation with a non-zero modulationstrength δε 6= δµ ≥ 0 exhibits horizontal β-band-gaps, whichoccur at the synchronization points between the forward andbackward STHs.

One may draw an analogy between a wave traveling inan STM medium and an airplane traveling through the air.Fig. 6(d) demonstrates the occurrence of three different caseswhen the velocity of an airplane changes with respect to thesound velocity. In the sonic (γ = 1) and supersonic (γ > 1)regimes, the velocity of the airplane is equal and larger thanthe velocity of the sound, respectively. In these cases, shockwaves start at the nose and end at the tail of the aircraft.The waves extend in different radial directions which resultin a cone similar to that of the plane’s vapor trail, called aMach cone, which is shown in the photo on the right side ofFig. 6(d). When the leading parabolic edge of the cone contactsthe earth, it creates a huge amount of sound energy, where theair particles are pushed aside with a huge force, and whichis experienced on land as a sonic boom. The pressure wavesgenerated in front of and behind the airplane travel at the speedof sound and vary by altitude and temperature. Hence, as theairplane travels faster and faster, the pressure waves cannotget out of the way of each other; instead they build up andcompress together and eventually form a single shock wave atthe speed of sound. The shape of the shock wave is called aMach cone and the opening angle of the cone is representedby sin(θcone) = vsound/vobject. A shock wave of light can alsobe generated. Light travels at the speed of 3× 108 m/s in theair but it slows down in natural materials by the factor of therefractive index of the material. Hence, a very fast particle canexceed the speed of light in a material with a refractive indexof larger than unity. For instance, a visible shock wave of lightcalled Cherenkov radiation can be produced by accelerating


electrons to a very high velocity and firing them into a pieceof glass or plastic.

It can be shown that the ST Bloch-Floquet decompositionin (4) is valid for subluminal (vm < vb) and superluminal(vm > vb) ST modulations. However, the ST Bloch-Floquetdecomposition is not convergent in the sonic interval [4], [11],[13], i.e.,

γs,min =

√εrµr

ε0µ0(1 + δε)(1 + δµ)≤ γ

≤√

εrµr

ε0µ0(1− δε)(1− δµ)= γs,max. (8)

where γs,min and γs,max are the lower and upper sides of thesonic interval, respectively. The sonic condition correspondsto the luminal ST modulation where the modulation andbackground velocities are close. This is due to the fact thatnone of the amplitudes of the infinite number of ST harmonicsare small in the sonic condition [4], [11], [13].

C. Numerical Simulation

Numerical simulation of an STM medium is an excellenttool for the parametric investigation of wave propagation andtransmission through such a medium, especially in the sonicinterval where the Block-Floquet solution does not converge.Fig. 7(a) sketches the implemented finite-difference time-domain (FDTD) scheme for numerical simulation of generalST permittivity- and permeability-modulated media [13]. Inthis scheme, the STM medium is sampled with K + 1 spatialsteps and M +1 temporal steps, with the steps of ∆z and ∆t,respectively. Then, the finite-difference discretized form of theelectric and magnetic fields may be achieved by satisfying thesourceless Maxwell equations at each ST sample, as [13]

Hy|i+1/2j+1/2=

1−∆tµ′|i−1/2j+1/2

µ|ij+1/2

Hy|i−1/2j+1/2

− ∆t/∆z

µ|ij+1/2

(Ex|ij+1−Ex|ij

)(9a)

Ex|i+1j =

(1−

∆tε′|ijε|i+1/2j

)Ex|ij−

∆t/∆z

ε|i+1/2j

(Hy|i+1/2

j+1/2−Hy|i+1/2j−1/2

)(9b)

where µ′ = ∂µ(z, t)/∂t = ωmµ0µavδµ sin(βmz − ωmt) andε′ = ∂ε(z, t)/∂t = ωmε0εavδε sin(βmz − ωmt).

We next investigate the effect of ST velocity ratio γ = 1and the effect of the length of the STM slab L on the forwardtransmission. It should be noted that, due to the negligibleeffect of the ST modulation on the backward transmission,varying the γ = 1 and L leads to minor change on thebackward problem. Fig. 7(b) plots the FDTD numerical resultsfor the amplitude of the electric field of the STHs at the outputof the STM slab versus the ST velocity ratio γ. It may beobserved from this figure that, in the subsonic interval γ → 0,the energy resides mainly inside the fundamental harmonic(n = 0). However, in the sonic condition, a strong energytransition occurs from the fundamental harmonic to the higherSTHs.

Fig. 7(c) plots the amplitude of the transmitted harmonicsversus the length of the STM slab. The structure operates inthe middle of the sonic condition, i.e. γ = 1. As the lengthof the slab is increased, the input power is more efficientlycoupled to the ST harmonics ω0 ± nωm, n ≥ 1. The incidentwave gradually couples its energy to these harmonics as itpropagates through the slab. Therefore, a longer slab exhibits amore efficient power transition but in a quasi-periodic manner.It may be seen from Fig. 7(c) that a quasi-periodic transitionof power between the fundamental harmonic (n = 0) andhigher-order harmonics occurs. One may derive a closed formsolution for the period of the wave transition between the fun-damental STH and the first higher (or lower) STH by ignoringthe effect of all other STHs and considering a pure transitionfrom n = 0 to n = 1 [9]. Thereby, the coherency length lc isachieved, which depends on the modulation parameters, i.e.,ωm, ω0, γ, δε and δµ [9].

The factor ωm/ω0 specifies the frequency of the generatedSTHs, and therefore, it is a crucial parameter. However, it alsodirectly affects the amplitude of the generated and transmittedSTHs. Fig. 8 presents the numerical simulation results thatinvestigate the effect of the modulation temporal frequencyωm/ω0 on the transmission gain, where δε = 0.22, δµ = 0,L = 3λ and γ = 1. It may be seen from this figure thata transmission gain is achieved for ωm/ω0 > 1. This isconsistent with the results presented in [43]. Moreover, ina space-time-varying system, the ratio of the power of thegenerated harmonics is Pn/P0 = |βn/β0| [34], where Pn andP0 are powers of the nth and the fundamental (n = 0) STHs,respectively [34]. This phenomenon has also been reported inthe Manley-Rowe relation for a time-varying medium [44].

D. Nonreciprocal Scattering

We next demonstrate the effect of unidirectionality of the STmodulation by investigating the wave transmissions throughthe STM slab for forward and backward problems. Figs. 9(a)and 9(c) plot, respectively, the time-domain and frequency-domain numerical result for the amplitude of the electric fieldin the forward problem. Here, the incident wave impinges onthe quasisonic STM slab (γ = 0.85, with γs,min = 0.867)possessing sinusoidal STM permittivity of δε = 0.3 anduniform permeability δµ = 0. It may be observed fromthese figures that the wave is strongly interacting with themedium, the incident power at ω0 is effectively transferredto the STHs at ω0 ± nωm, n ≥ 1, leaving a weak transmittedwave at the incident frequency ω0. In contrast, and the incidentwave in the backward problem (Figs. 9(b) and 9(d)) passesthrough the STM slab with weak interaction and minor powerexchange with the ST modulation. These plots reveal that theST medium behaves differently under forward and backwardwave incidences, i.e., providing strong STHs in the forwardproblem, but weak STHs in the backward problem.

E. Superior Performance of Equilibrated ST Modulation

It has been shown that an equilibrated STM medium(δµ = δε > 0) presents much stronger nonreciprocity incomparison with conventional permittivity STM medium [13].


mω

00

0ω

mβ

0β

light line

0n=

-1n=

+1

n=

0n=

-1n=

β −∆

β +∆

Slope=γ

Subluminal ST modulation, cmv <

(a)

0ω

mω

00

0ω

mβ

0β

-Nn= +N

n=0

n=

-1n=β −

∆ β +∆ ×(2 1)N +

Luminal ST modulation, cmv =

Overlapping

(b)

mω

00

0ω

mβ

0β

0n=

-1n=

+1

n=0

n=

-1n=

+1

n=

Superluminal ST modulation, cmv <

(c)

Shock cone

Overlapping

Subsonic Sonic Supersonic Shock waves

in a supersonic airplane

<, =, >, jetv soundvsoundvjetvsoundvjetv

soundvjetv <

β +∆β −

∆

(d)

Fig. 6. Analogy between a sonic airplane and an STM medium. (a)-(c) Normal incidence dispersion diagrams for the sinusoidally STM (unbounded) slabmedium. (a) Subluminal ST modulation with δε = 0.4, δµ = 0 and γ = 0.5. (b) Same as (a) except for the vanishingly small permittivity modulationamplitude δε = δµ → 0 and the subsonic ST velocity ratio γ → 1. (c) Superluminal ST modulation with δε = 0.4, δµ = 0 and superluminal ST velocityratio γ = 3.5. (d) Sonic airplane travels at three different speeds, i.e., subsonic, sonic and supersonic [33].

Such a strong nonreciprocity is accompanied by a larger soniccondition interval which provides extra design freedom forachieving strong nonreciprocity by a weak pumping strength.It is shown that the width of photonic band gaps in generalperiodic space-time permittivity- and permeability-modulatedmedia is proportional to the absolute difference between theelectric and magnetic pumping strengths |δε − δµ|. Con-ventional space-time permittivity-modulated media providemoderate photonic transitions from the excited mode to itstwo adjacent modes. However, in an equilibrated space-time-varying medium, strong photonic transitions occur from theexcited mode to its four adjacent modes [13].

Fig. 10(a) compares the dispersion diagrams of a non-equilibrated ST permittivity-modulated medium with δε = 0.2,δµ = 0 and γ = 0.8, with the dispersion diagram ofan equilibrated ST permittivity- and permeability-modulatedmedium with δε = δµ = 0.2 and γ = 0.8. It is obviousfrom this figure that, the conventional, non-equilibrated, STMmedium exhibits photonic band gaps at the ST synchronizationpoints, i.e., at the intersections of the forward and backwardSTHs which lie on oblique lines. Such band gaps correspondto evanescent waves, where the wave propagates throughthe medium, and experiences high attenuation and reflection.Nevertheless, by switching on the permeability modulationwith the same ST modulation amplitude of δµ = δε = 0.2,

equilibrium occurs in the electric and magnetic properties ofthe medium, and photonic band gaps are closed, for all valuesof γ [13]. Another interesting phenomenon that we see inFig. 10(a) is as follows. The equilibrium in the medium hasled to significant enhancement in the nonreciprocity of themedium, so that the equilibrated medium introduces muchstronger nonreciprocity, ∆β+(δµ = 0.2) << ∆β+(δµ = 0).The nonreciprocity (NR) of the STM medium may also bedefined in the frequency domain for the nth harmonic, asthe ratio of the forward transmitted field to the backwardtransmitted field, i.e., NRn = EF

T,n/EBT,n.

Fig. 10(b) plots the nonreciprocity enhancement providedby the equilibrated STM medium (δε = 0.02, δµ = 0) incomparison with the non-equilibrated STM medium (δµ =δε = 0.02), for ω0 = 2π × 1.5 GHz, ωp = 2π × 0.2 GHz,L = 20λ0 and γ = 1. As we see from this figure, theequilibrium in the electric and magnetic properties of the STMslab has led to significant enhancement in the nonreciprocityof STHs. In particular, the nonreciprocity of the fundamentalSTH, which is the most important harmonic to be largelyisolated [11], [24], [34], has enhanced more than 360%, andthe nonreciprocity of most of the STHs has enhanced morethan 220%.

Since the constitutive parameters of an STM medium varyin both space and time, the intrinsic impedance of the medium


z

z

z

z∆ z2∆0 ( 1)K z+ ∆1j = 2j = 3j = 1j K= +

0t =

t t= ∆

/ 2t t= ∆

0i =

1/2i =

1i =

xE

xE x

ExE

xE

xE x

ExE

yH

yH

yH y

H

zi = 1M +

xE

xE x

ExE

( 1)t M t= + ∆

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

FF

TI

/E

E

γ

0n =

+1

n= -1

n=

+2n =

-2

n= +3

n= -3n =

sonic region

(b)

(c)

Fig. 7. Finite-difference time-domain numerical simulation of the STpermittivity- and permeability-modulated media. (a) Space-time samplingscheme [13]. (b) and (c) Numerical simulation results for harmonic generationin non-equilibrated STM medium with parameters δε = 0.22, δµ = 0,ωm = 2π × 0.2 GHz and ω0 = 2π × 1.5 GHz versus (b) the velocityratio γ and (c) the length of the slab for γ = 1 [11].

0 1 2 30

0.5

1

1.5

2

2.5

FF

TI

/E

E

0

+3

-1

+1

-2

+2

-3

mω

0ω/

n =

Fig. 8. Investigation of effect of the modulation temporal frequency ωm/ω0on the transmission gain, based on the numerical simulation results, for δε =0.22, δµ = 0, L = 3λ and γ = 1.

depends on both space and time, i.e.,

η(z, t) =

√µ(z, t)

ε(z, t)=

√µ0µav[1 + δµ cos(βmz − ωmt)]

ε0εav[1 + δε cos(βmz − ωmt)](10)

which represents an ST-varying intrinsic impedance. For δµ 6=δε, (10) corresponds to a non-equilibrated STM medium withlocal ST reflections (R(z, t)) in an infinite number of spacepoints, i.e., R(z0, t) = [η(z0 + ∆z, t) − η(z0, t)]/[η(z0 +∆z, t)+η(z0, t)] and in an infinite number of time points, i.e.,R(z, t0) = [η(z, t0 +∆t)−η(z, t0)]/[η(z, t0 +∆t)+η(z, t0 +∆t)]. This issue is overcome by an equilibrated STM medium(δµ = δε), where the ST-varying intrinsic impedance in (10)reduces to

η(z, t)∣∣δε=δµ

= ηeq = η0ηav (11)

which is constant and independent of both space and time,yielding zero local ST reflections. Figs. 10(c) and 10(c) com-pare the wave reflections (ER) in non-equilibrated and equi-librated STM media, respectively for forward and backwardproblems. It is obvious that the conventional non-equilibratedSTM slab exhibits strong space and time local reflections,whereas the equilibrated STM slab presents no space and timelocal reflections.

Fig. 10(e) illustrates two circuit models for sinusoidallyST permittivity- and permeability-modulated slabs. These cir-cuits are made up of an array of subwavelength-spaced unit-cells, each of which is composed of a variable inductoror variable mutual inductance and a variable capacitor ora variable mutual capacitance. These variable elements areplaced, respectively, in series and parallel with the intrinsicinductance and capacitance of the transmission line, andare spatiotemporally modulated using a sinusoidal harmonicwave, i.e., VP cos(ωmt). Interestingly, such a structure withhighly enhanced nonreciprocity, shown in Fig. 10(b), requiresthe same pumped energy as a conventional ST permittivity-modulated structure [10], [11], [43].


−1.5

−1

−0.5

0

0.5

1

1.5

30 6 9 12-3-6-9

0z

F()

Ez

0( )z λ

(a)

0 0.5 1 1.5 2 2.5 3−40

−30

−20

−10

0

0 0.5 1 1.5 2 2.5 3−40

−30

−20

−10

0

Freq.(GHz)Freq.(GHz)

F I(dBW)

E

F T(dBW)

E

Anal.Num.

(b)

−1.5

−1

−0.5

0

0.5

1

1.5

B()

Ez

1z

30 6 9 12-3-6-9

0( )z λ

(c)

0 0.5 1 1.5 2 2.5 3−40

−30

−20

−10

0

0 0.5 1 1.5 2 2.5 3−40

−30

−20

−10

0

Freq.(GHz) Freq.(GHz)

B I(dBW)

E

B T(dBW)

E

Anal.

Num.

(d)

Fig. 9. Analytical and numerical (FDTD) results for the forward and backward problems in the quasisonic condition with parameters δε = 0.3, ω0 =2π × 1.5 GHz, ωm = 2π × 0.2 GHz, L = 3λ0 and γ = 0.85 [11]. (a) and (c) FDTD waveforms showing the electric field amplitude for the forwardand backward problems, respectively [11]. (b) and (d) Frequency spectrum of the incident and transmitted fields for the forward and backward problems,respectively [11].

IV. APPLICATIONS OF ST MODULATION

Recently, ST modulation has begun to be used for therealization of a variety of novel and efficient components. Thissection presents a class of guided and radiated structures thatleverage the unique and valuable properties of STM media.

A. Magnet-Free Linear Isolators

1) Quasisonic Isolator: The strong nonreciprocity of thequasisonic condition is utilized for the realization of a mi-crowave isolator [11]. The length (L) and the modulationratio (γ) are adjusted such that in the forward direction theincident power is efficiently transferred to higher order STHsω0 ± nωm, with n ≥ 1, whereas a small amount of power istransmitted at the fundamental frequency ω0. In contrast, inthe backward direction the STM slab interacts weakly withthe incident wave and therefore passes through the slab nearlyunaltered. Fig. 11(a) shows the architecture of the designedquasisonic isolator and the experimental results. In the forwardproblem, the transmitted wave passes through a bandpass filter(BPF in the figure), with bandpass frequency ω0, and sincemost of the power transitted to higher order STHs will bereflected back, as it lies in the the stopband of the filter.However, in the backward direction most of the power isresiding at the fundamental frequency ω0, and hence, passesthrough the slab with minimal alteration. Thus, the structureoperates as an isolator at frequency ω0. The microwaveimplementation of this isolator is carried out by emulating

the STM medium with the constitutive parameters in (6), withδµ = 0, by a microstrip transmission line loaded with an arrayof subwavelength-spaced varactors. The fabricated prototypeis shown in Fig. 11(b), where the varactors are reverse-biasedby a DC voltage and are spatiotemporally modulated by anRF bias. Such a structure realizes the STM capacitance

C(z, t) = Cav[1 + δC cos(βmz − ωmt)]. (12)

where Cav is proportional to the effective permittivity of themicrostrip line and δC corresponds to the average permittivityintroduced by varactors. The ST modulation amplitude maybe controlled via the amplitude of the modulation RF bias.The experimental results for the electric field amplitude ofthe incident, transmitted and reflected waves are plotted inFig. 11(a). In the forward direction, corresponding to the topof Fig. 11(a), the transmission level is less than −20 dB atthe fundamental harmonic, and less than −30 dB in otherST harmonics, while the transitted power to higher orderharmonics ω0±nωm is reflected by the bandpass filter. In thebackward direction, shown in the bottom of Fig. 11(a), theincident wave is nearly fully transmitted at the fundamentalfrequency, with a weak transition to higher order STHs withless than −30 dB reflection. Hence, the medium provides morethan 20 dB isolation. The insertion loss of the passing direction(backward direction) is about 1.9 dB, of which 1.2 dB is dueto the bandpass filter insertion loss.

2) Isolator Based on Asymmetric Photonic Band Gaps: Re-ciprocal media support perfectly horizontal photonic band gaps


0.7 0.75 0.8 0.85 0.9 0.95

1.1

1.2

1.3

1.4

0 p/β β

0p

/ω

ω

Ban

dg

aps

for

µδ

0=

0.2,εδ = 0µδ = εδ = 0.2µδ =

β +∆ , µδ 0=

β +∆ , µδ 0.2=

(a)

-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7

1

2

5

10

20

50

n

Nonre

cipro

city

enhan

cem

ent

(100%

)

(b)

0

0.2

0.4

0.6

0.8

1

1.2

0z0z

F(

)||E

z

F

IE

RE

0 0,ε µ

0.5εδ =0,µδ =

F

0

0.2

0.4

0.6

0.8

1

1.2

0z0z

F(

)||E

z

F

IE

RE

0 0,ε µ

0.5εδ =µδ =

F

(c)

0

0.2

0.4

0.6

0.8

1

1.2

z 0z

B(

)||E

z

RE

0 0,ε µ

0.5εδ =0,µδ =

L

B

IE

B

0

0.2

0.4

0.6

0.8

1

1.2

z 0z

B(

)||E

z

B

IE

RE

0 0,ε µ

L

0.5εδ =µδ =

B

(d)

x

z

m m mcos( )V z tβ ω−

z∆

r[1 cosµµ δ+r[1 cosεε δ+

m m( )]z tβ ω−

m m( )]z tβ ω−

0L varL

0C varC

(z,t)

(z,t) 0,varC0C

varL (z,t)0L

m mcos( )V tω

x

zz∆

2L

L

1L

mCm

2C

1C

(z,t) (z,t)

2L

L

1L

mCm

2C

1C

(z,t) (z,t)

2L

L

1L

mCm

2C

1C

(z,t) (z,t)

m m mcos( )V z tβ ω−

(e)

Fig. 10. Equilibrated sinusoidally STM medium. (a) Dispersion diagram comparison between the equilibrated (δµ = δε = 0.2, γs,l = 0.833) and non-equilibrated (δµ = 0, δε = 0.2, γs,l = 0.913) STM media with γ = 0.8 [13]. (b) Nonreciprocity enhancement of the equilibrated ST slab in comparison withthe nonreciprocity provided by the conventional ST slabs [13]. (c) and (d) Reflection from conventional and equilibrated STM slabs for the forward problem(c) and the backward problem (d) [13]. (c) Circuit model for an equilibrated STM transmission line [13]. (e) Circuit model for an equilibrated STM coupledtransmission line [13].

Fig. 11. Quasisonic STM isolator. (a) Experimental results for the forward and backward problems [11]. (b) Circuit model and a photograph of the fabricatedisolator [11].


in the dispersion diagram, which are symmetric with respect tothe positive and negative Bloch-Floquet wavenumbers. In theband gaps, the Bloch-Floquet harmonics acquire an imaginarypart in their wavenumber and hence become evanescent. Thus,when a wave incident on the structure is modulated at afrequency falling within these band gaps, complex and henceevanescent modes will be excited. In contrast to reciprocalmedia, STM media provide oblique ST photonic transitionsand exhibit asymmetric band gaps [11], [45]. Fig. 12(a)depicts the wave isolation based on asymmetric photonicband gaps [45]. This approach leverages the ST variationin the permittivity of medium to generate electromagneticband structures that are asymmetrically aligned with respectto the direction of propagation, as shown in Fig. 12(b). Inthe forward problem, the incident wave at the frequency ω0

excites an evanescent mode, represented by the magenta dot.If the structure is long enough, almost no power reachesthe end of it and the wave is fully reflected. In contrast,in the backward problem, the mode marked by the greendot is a propagating mode, which is excited. Therefore, theincident electromagnetic power is transferred to the other sideof the structure. Fig. 12(c) shows an image of the fabricatedprototype and the experimental results for the forward andbackward transmitted fields. The ST modulation circuit isformed by 39 unit cells of antiparallel varactors, uniformlydistributed along the microstrip line, with the subwavelengthperiod p = 5 mm, corresponding to p/λm ≈ 1/19 [45]. Hence,the structure emulates a medium with the continuous STMpermittivity in (3a). The corresponding length at frequencyω0 = 2π×2.5 GHz is L = 6λ0, where ωm = 2π×0.675 GHz,βm = 415.79 rad/m, and δC = 0.15. The achieved isolation atω0 = 2π × 2.5 is more than 10.5 dB.

3) Self-Biased Broadband Isolator Based on UnidirectionalST Coherency: Consider a condition in which the STmodulation and the incident wave share the same temporalfrequency ω0 [18]. As a consequence, a temporal coherencyoccurs between them regardless of the direction of the inci-dent wave, and the nonreciprocal operation of the structureis dictated by the spatial coherency difference between thedirection of the STM medium and the incident wave. At certainST modulation phase shifts and ST modulation amplitudes,corresponding to the ST coherency conditions, the structureoperates as an isolator. The proposed structure provides broadoperation bandwidth and small size, and hence, exhibitssuperior efficiency compared to previously proposed STMisolators [9]–[11], [45]–[47]. Such a coherent STM isolatormay be realized using a self-biased architecture, where theinput signal modulates the structure itself, and it thus operatesas a self-biased isolator. Moreover, the proposed isolator iscapable of providing transmission gain, as well as introducingnonreciprocal reflection gain. This isolator is formed by atransmission line with the length L and permittivity

ε(z, t) = εav[1 + δε cos(βmz − ω0t+ φm)]. (13)

where φm is the modulation phase, and ω0 denotes the tem-poral frequency of both the modulation and incident wave.The closed form solution of the scattered electromagneticfields may be derived, by considering the transformation

Fig. 12. Isolator based on nonreciprocal Bragg reflections in an STMmedium with asymmetric electromagnetic band gaps. (a) Forward problem:propagation of the wave in the electromagnetic band gap of the structureyields complete-reflection. Backward problem: propagation of the wave inthe pass-band of the structure yields complete-transmission. (b) Dispersiondiagram [45]. (c) Photograph of the fabricated structure and the experimentalresults for nonreciprocal transmission [45].


W = γ√

8δεej(βmz−ω0t+φm)/2 and t′ = t. Then, the electric

field inside the spatiotemporally coherent medium may beexpressed in terms of the modified Bessel functions of thefirst and second kinds [18].

Fig. 13(a) provides a generic representation of nonreciprocalelectromagnetic wave transmission through the unidirection-ally perfectly coherent STM medium, with perfect forwardcoherency and perfect backward incoherency. Fig. 13(b) plotsthe closed-form solution results for the forward and backwardtransmissions versus the modulation phase φm for βm = β0,δε = 0.15, εav = 7.06, ω0 = 2π × 2 GHz and withL = 2λg yielding a maximal contrast between the forwardand backward transmissions. However, to achieve a strongerisolation level and arbitrary transmission gain, we considerimperfect space-coherency, where βm 6= β0 (γ 6= 1), andperfect temporal coherency. Fig. 13(c) shows that, forwardtransmission of EF

T/EI = 0 dB with the isolation level of 24.5dB may be achieved for γ = 6.66. A forward transmissionpeak is observed around γ = 3.063 which corresponds tothe perfect spatial coherency of the forward transmitted wave,where βmL = φm. Moreover, a reciprocal transmission occursat γ → ∞ (βm → 0) and γ → 0 (ω0,m → 0) [11], wherethe structure is undirected. Fig. 13(d) shows the architectureand an image of the fabricated self-biased isolator using threeWilkinson power dividers, extracting the ST modulation wavefrom the input wave. Fig. 13(e) plots the experimental andsimulation results for the forward and backward transmissionsthrough the spatiotemporally coherent isolator, where morethan 8 dB isolation (with linear phase response) is achievedacross 122% fractional bandwidth. To achieve a higher isola-tion level and stronger transmitted signal, one may enhancethe ST modulation amplitude δε by amplifying the extractedmodulation wave using a transistor-based amplifier. We shallstress that this device provides the same linear response for amodulated or non-sinusoidal input signal. The dynamic rangeof the isolator, for more than 15 dB isolation, is about 30dB, i.e., from −10 dBm to +20 dBm. There is a danger thatthe proposed solution for amplifying the extracted modulationusing a transistor-based amplifier, permitting operation withsmall input-signals, could compress the varactors and restrictthe maximum power handling and linearity. This could beprevented by using an automatic gain control (AGC) thatcontrols the gain of the amplifier.

B. STM Magnet-Free Circulator

Circulators are nonreciprocal three-port devices which arethe fundamental units of all full-duplex communication sys-tems. Assuming a clockwise circulator, the signal entering intoit (port 1) exits from the next port in a clockwise direction(port 2), where port 3 is isolated. However, if some of theexited light is reflected back to the circulator, it come outof port 3, where port 1 is isolated. Recently, there has beensubstantial scientific interest in wave circulation based onST modulation [21]–[23], [43], [48]. STM circulators exhibitsuperior performance over conventional magnetically biasedferromagnetic circulators and active circulators. An elegantapproach for the realization of an STM microwave circulator

would consist of STM coupled-resonators, where the effectivespin is imparted to the structure by the ST modulation [21],[48]. Such a structure is compatible with integrated circuits andcould potentially exhibit good power and noise performance.Fig. 14(a) shows a generic representation of the STM circula-tor presented in [21], formed by three identical time-modulatedresonators symmetrically coupled to three transmission lines.These three resonators are time-modulated so that the time-dependent resonance frequency of the ith resonator reads

ωi(t) = ω0 + δωm cos

(ωmt+ (i− 1)

2π

3

), i = 1, 2, 3

(14)where ω0 is the resonance frequency of the unnmodulated res-onators, ωm is the modulation frequency and δωm is the ampli-tude of the resonance-frequency perturbation [21]. Therefore,three identical resonators are time-modulated with the samefrequency and amplitude, but phases of 0◦, 120◦ and 240◦. Inthe absence of the time modulation, the loop of resonatorssupports two degenerate counter-rotating modes possessingphases 0◦, ±120◦ and ±240◦ in the three resonators. Thetime modulation of resonators results in right-handed and left-handed states, so that exciting the circulator from, for instance,port 1 yields excitation of the right- and left-handed stateswith the same amplitude and opposite phases φR = −φL.It can be shown that by exciting port 1, the signal at port2 reads ei2π/3eiφR + e−i2π/3eiφL and the signal at port 3is ei4π/3eiφR + e−i4π/3eiφL [21], [48]. Hence, by choosingφR = π/6 and φL = −π/6 a constructive interference atport 2 and destructive interference at port 3 is achieved,yielding an appropriate signal circulation. Fig. 14(b) shows theexperimental realization of this STM circulator at microwavefrequencies based on a loop of lumped LC resonators, varac-tors and wye resonators. Fig. 14(c) shows a photograph of thefabricated prototype, and Fig. 14(d) plots the simulation andexperimental results. The microwave version of this circulatormay be realized using distributed elements. In addition, [21]presents a microwave version of the proposed circulator basedon distributed elements for wireless-communications band 2.2GHz, where theoretical and simulation results are presented.

C. STM Transceiver Front-Ends

Thanks to the extraordinary properties of STM media,various multifunctionality components have been recently re-ported. These components are appropriate for the front-endof RF communication systems, where a single STM structuresimultaneously may provide the tasks of frequency up-/down-conversion, duplexing, antenna and filtering process.

1) Broadband low-loss nonreciprocal component:Fig. 15(a) [20] shows an STM structure which is designedfor broadband and low-loss nonreciprocal signal transmissionand reception. The operation of this device is as follows.The temporal frequency of the ST modulation is largerthan that of the input signal, i.e., ωm > ω0. In the transmitmode (left-to-right, backward problem), the input signal atω0 passes through the structure with nearly no alteration inthe frequency and amplitude. However, in the receive mode(right-to-left, forward problem), the incoming signal from the


Perfect coherency

Perfect incoherency

Backward problem

Forward problem

F FT IE E>F

IE

BIE

BTE 0≈

z

L0

g / 2L λ=

Space-time modulated medium

av m m[1 cos( )]z tεε δ β ω π+ − +

(a)

-3

-2

-1

0

1

2

3

Isola

tion

Backward Forward

π 2π2π−3π− π− 0 3π

TI

Tra

nsm

issi

on

,/

(d

B)

EE

mφ

mLβm=β 0β m=v,

bv

(b)

(c)

Port 1 Port 2

FI 0E ,ω F

T 0E ,ω

BI 0E ,ω

BTE 0≈

Power

Power Powerdivider

divider

divider

Mod.

z

R

Port 1Port 2

Powerdivider

av m m m[1 cos( )]z tεε δ β ω φ+ − +

(d)

0 0.5 1 1.5 2 2.5 3 3.5 4−35

−30

−25

−20

−15

−10

−5

0

TI

Tra

nsm

issi

on

,/

+3 (

dB

)E

E

Freq.(GHz)

Forward-simulation

Backward-simulation

Forward-measurement

Backward-measurement

(e)

Fig. 13. One-way ST coherency in a STM medium, i.e. ωm = ω0. (a) Unidirectional perfect space-and-time coherence, i.e., βm = β0 [18]. (b) Effect ofthe modulation phase, φm, on the forward and backward transmissions with δε = 0.15 [18]. (c) Transmission versus the ST velocity ratio [18]. (d) Circuitalschematic and photograph of the fabricated structure [18]. (e) Simulation and experimental results [18].


(a) (b)

(c) (d)

Fig. 14. Magnetless circulator based on time-modulated coupled resonators. (a) Generic schematic of the circulator [21]. (b) Realization using lumped-elementvaractor-based wye resonator. The leakage of the modulation signal to the external ports is prevented using three parallel bandstop filters (Lbsf and Cbsf) [21].(c) Photograph of the fabricated prototype [21]. (d) Experimental and numerical simulation results [21].

antenna at ω0 is up-converted to ω−1 = ωm − ω0. Hence,this device can be used as a circulator if the TX/RX port isconnected to a diplexer, with two pass bands at ω0 and ω−1separating the TX and RX paths. In addition, to cancel outthe up-conversion in the down-link, an external mixer maybe used in the RX port side for down-conversion from ω−1to ω0. The complete analysis of the structure and its noiseperformance is provided in [20].

This nonreciprocal component is designed at microwavefrequencies using two pairs of differential transmission linesloaded with reverse-biased varactors in a double-balancedfashion. They support separate traveling of the modulationsignal and the main signal, where C+ and C− denote themodulation (carrier) transmission lines, and S+ and S−represent the main signal transmission lines. Due to thesymmetry, these two transmission-line pairs are isolatedfrom each other, where each transmission line pair is onthe virtual ground of the other one [20]. Fig. 15(b) plotsthe theoretical, simulation and experimental results for theisolation (S31), transmission gain (S21), and receive gain(S32) [20] provided by the nonreciprocal component in

Fig. 15(a). The results demonstrate a broadband operation,that is, from ω0 = 2π × 0.4 GHz to 2π × 1.8 GHz.

2) Nonreciprocal transmitter: The ST modulation may berealized in conjugation with the leaky-wave modes. As a con-sequence, in addition to the frequency mixing and nonrecipro-cal wave transmission, a leaky-wave radiation will be enabled,leading to nonreciprocal mixer-antenna structures [29], [30],[34]. The ST modulation is well-integrated with the leaky-wave structures since both the ST modulation and leaky-waveradiation are based on progressive change in the travelingwave. In particular, the STM leaky-wave structure has beenproposed for the front-end of transmitters providing frequencyup-conversion and radiation [29], [30]. Fig. 16(a) depicts theapplication of the STM leaky-wave medium to the front-end of a transmitter [30], where a desired up-conversion andleaky-wave radiation is achieved in the transmit mode. In thisscheme, the radiation is enabled by periodic interruption (bythin radiation apertures slots) in a coplanar waveguide (CPW).The voltage-dependent capacitors are placed right below theslots to control the propagation phase. The modulation controlis achieved using a diplexer that combines the main RF signal


Fig. 15. ST modulation yielding broadband and low-loss signal circulation. (a) Circuital representation and an image of the fabricated prototype. In thedown-link, up-conversion from ω0 to ω−1 = ωm − ω0 occurs, so that additional down-conversion to ω0 using an external mixer is required [20]. (b)Theoretical, simulation and experimental results for isolation (S31), transmission gain (S21), and receive gain (S32) [20].

and the modulation signal in a single port connected to theantenna via a Bias-T to superimpose the DC bias [30].

3) Full-duplex transceiver: The ST modulation has beenintegrated with the leaky-wave medium in [34], for the re-alization of a full transceiver front-end, composed of up-converter, duplexer, leaky-wave antenna and down-converter.Fig. 17(a) shows the operation principle of the STM mixer-duplexer-antenna leaky-wave structure. Such a structure op-erates based on the up-link and down-link ST transitions(shown in Fig. 17(a)), as up-conversion and down-conversionmixing operations, and separation of the up-link and down-link paths as a duplexing operation. Moreover, it provides STfrequency beam scanning at fixed input frequency by virtue ofthe modulation parameters (here the modulation frequency).In the up-link, a signal with frequency ω0 is injected intothe transmitter port and propagates in the +z direction, corre-sponding to the right-hand side of the dispersion diagram inFig. 17(a). As a result of mixing with the periodic modulation,this signal experiences progressive temporal frequency up-conversion from ω0 to ω1 = ω0 + ωm along with spatialfrequency transition from β0 to β1 = β0 + βm.

In the down-link, the signal with frequency ω1 is impingingon the structure under the same angle θ1 and picked up by thestructure. In conformity with the phase matching, the signalcan only propagate in the direction of the modulation, i.e., inthe +z direction, corresponding to the right-hand side of the

dispersion diagram in Fig. 17(a), and experiences progressivespatial and temporal frequency transitions (down-conversion),from β1 to β0 = β1 − βm and from ω1 to ω0 = ω1 − ωm,respectively, while propagating toward the receiver port. NoST transition is allowed for signal propagation in the backward(−z) direction (left-hand side of the dispersion diagram inFig. 17(a)) due to the unavailability of leaky modes at thepoints (−β0 +βm, ω0 +ωm) from (−β0, ω0) for up-conversionand (−β1−βm, ω0) from (−β1, ω0+ωm) for down-conversion.The radiation angle (for both transmit and receive at frequencyω1) is denoted by θ1, which is given by [34]

θ1 = sin−1(β1(ω)

k01

)= sin−1

(c(β0 + βm)

ω0 + ωm

), (15)

where k01 = ω1/c is the effective wavenumber at the fre-quency ω1. According to Eq. (15), the radiation angle dependson both input and modulation frequencies, ω0 and ωm. Hence,we take advantage of this property and achieve space scanningby varying the modulation frequency, i.e., θ1 (ωm), where theinput frequency ω0 is fixed. Beam scanning can be achieved ata fixed input frequency, ω0, simply by varying the modulationfrequency, ωm.

Fig. 17(b) shows the details of the fabricated prototype. Theantenna system supports two modes, i.e., the narrow strip cor-responds to a dominant even mode, which provides a guided-mode path for the RF signal modulating the varactors, and thewide strip which represents a higher order odd (EH1) mode,


(a)

(b)

Fig. 16. STM leaky-wave medium casting as a nonreciprocal mixer-antenna. (a) Circuit representation and an image of the fabricated prototype [30].(b) Experimental results for (left-side) Unmodulated structure, fm = 0, and (right-side) Modulated structure, fm = 600 MHz [30].

providing the leaky-mode channel for radiation. Fig. 17(c)plots the experimental result for the normalized radiated powerof the up-link frequency beam scanning. The input frequencyis set at f0 = 1.7 GHz, where the modulation frequencyset reads fm = {0.18, 0.22, 0.27, 0.3} GHz, correspondingto radiation frequencies f1 = {1.88, 1.92, 1.97, 2} GHz andradiation beam angles of θ1 = {4, 11.5, 18, 24.5}◦. Figs. 17(d)and 17(e) plot the experimental results for the normalizedreceived power at the receive and transmit ports for down-linkfrequency beam scanning. The input frequency at the refer-ence antenna port varies as f1 = {1.88, 1.92, 1.97, 2} GHz,corresponding to fm = {0.18, 0.22, 0.27, 0.3} GHz, for afixed received signal f0 = f1 − fm = 1.7 GHz. Theisolation achieved at specified radiation angles corresponds to{31.5, 12.3, 9.2, 15.5} dB.

D. Pure Frequency Mixer Based on Aperiodic ST modulation

Frequency mixing is the key operation in telecommunicationcircuits and systems which is conventionally achieved bynonlinear components, e.g., diodes and transistors, or by lineartime-varying circuits where the nonlinearity is due to theparasitic effects of the diode or switch elements. The nonlinearresponse of these nonlinear components results in generation

of an infinite number of undesired mixing products. PeriodicST modulation provides the required energy and momentumfor transition from the fundamental temporal frequency to aninfinite number of ST frequency harmonics [4], [11], [49]. Ageneral technique may be used for pure frequency mixing inSTM media, where harmonic photonic transitions in tempo-rally periodic systems are prohibited by tailored photonic bandgaps introduced by the engineered spatial-aperiodicity of thestructure. In contrast to conventional periodic uniform STMmedia, here the medium is aperiodic in space and thereforeexhibits an aperiodic dispersion diagram and photonic bandgaps tailored in a way to provide photonic band gaps atundesired ST mixing products, and, hence, introduces a purefrequency mixing. In addition, the frequency bands of themixer may be tuned via ST modulation parameters. Theproposed mixer takes advantage of the aperiodic spatiallyvarying structures which are capable of providing a varietyof electromagnetic responses that are unattainable by uniformstructures. Such structures may be used for the realization ofhighly efficient electromagnetic systems [50]–[64].

Fig. 18(a) shows the schematic of an aperiodic STMpure frequency mixer, based on a nonuniform conductor-backed coplanar waveguide (CPW) loaded with an array ofsubwavelength-spaced varactors. The structure assumes the


Fig. 17. STM mixer-duplexer-antenna leaky-wave system. (a) STM leaky-wave medium as a full transceiver composed of up-converter, duplexer, leaky-waveantenna and down-converter [34]. (b) Realization of the space-time modulated mixer-duplexer-antenna system based on the half-wavelength microstrip leaky-wave antenna [34]. (c),(d),(e) Experimental results for the scanning at θ1 = {4, 11.5, 18, 24.5}◦ corresponding to f1 = {1.88, 1.92, 1.97, 2} GHz, i.e.,(c) up-link radiated power, (d) down-link received power at the RX port, and (e) down-link received power at the TX port [34].


thickness L, a spatially aperiodic temporally periodic permit-tivity, i.e.,

ε(z, t;ω) = ε0εap(z;ω) [1 + δm cos(βm,ap(z;ω)z − ωmt)] ,(16a)

and a spatially aperiodic permeability

µ(z;ω) = µ0µap(z;ω), (16b)

where εap(z;ω) and µap(z;ω) are aperiodic functions of spaceand frequency, and βm,ap(z) and ωm respectively denote thespatial frequency and temporal frequency of the pump wave.Such an STM mixer is represented by average spatially variantphase velocity vap(z;ω) = c/

√εap(z;ω)µap(z;ω); the pump

wave shows an aperiodic space-dependent spatial frequency ofβm,ap(z;ω) = ωm/vap(z;ω), where the subscript “ap” denotes“aperiodic” spatial variation of these functions. The input waveat ω0 is injected into the aperiodic STM medium in Fig. 18(a),and then, propagates inside the medium along the +z directionand experiences a photonic transition to ωn = ω0 + nωm.However, with a proper spatial-aperiodicity all the undesiredST mixing products can be significantly suppressed by virtueof aperiodic photonic band gaps, leading to a pure transitionfrom ω0 to ω1 = ω0 + ωm.

The spatial variation of the constitutive parameters, εap(z;ω)and µap(z;ω), is designed through a synthesis method whichprovides the specified dispersion diagram with proper photonicband gaps, so that |S21(ω0, ωm, ω1)| = 1 and |S21(ω <ω0, ω0 < ω < ωm, ω1 < ω)| → 0. The synthesis procedure,shown in Fig. 18(b), is based on optimization of the nonuni-form spatial profile of the CPW and the average permittivityof the varactors, εvar. In Fig. 18(b), W represents the widthof the middle line, s is the gap between the middle lineand the grounding plane and h denotes the height of thesubstrate. Figs. 18(c) and 18(d) plot the optimized analyticaldispersion diagram, i.e., the real and imaginary parts of thewavenumber Re{β(ω)} and Im{β(ω)}, for the unboundedaperiodic dispersion-tailored STM mixer with the optimizedpermittivity and permeability. It may be seen from thesetwo figures that periodic transitions to undesired STHs areprohibited (Fig. 18(c)) by virtue of aperiodic photonic bandgaps, where ∂Re{β(ω)}/∂ω → 0 and Im{β(ω)} >> 0.Thanks to the tailored spatial aperiodicity of the structure,except ω0 and ω1, all the time harmonics lie in the band gap ofthe structure and hence will not mature, but rather propagateas weak waves and are attenuated. Figs. 18(c) and 18(d) showthe suppression level at different band gaps for undesiredharmonics. It should be noted that, except for the input waveat ω0, the power level of all the space-time harmonics is equalto zero at the input of the structure, i.e., at z = 0. As aresult, the band gaps are to prohibit growth of the power ofthe undesired harmonics, rather than attenuation of the strongwaves. Consequently, even a weak or moderate attenuation in aband gap may lead to strong suppression of the correspondingspace-time harmonic.

Fig. 19(a) shows the experimental implementation of theaperiodic STM waveguide mixer with constitutive parametersin (16), considering the experimental set-up in Fig. 19(b).Fig. 19(c) plots the experimental results for the output spectra

of the periodic and aperiodic STM frequency mixers. Thisplot shows that the aperiodic dispersion-tailored STM mixerhas significantly suppressed undesired STHs, while providingthe required energy and momentum for a pure transition fromthe input frequency ω0 to the desirable harmonic at ω0 +ωm.

V. OUTLOOK AND CONCLUSION

Space-time-modulated (STM) structures represent anemerging class of dynamic and directional materials. STmodulation is a promising new paradigm for nonreciprocityand frequency generation, ranging from acoustics and radiofrequencies to terahertz and optics, and is compatible withhigh frequency techniques [10], [19]. This paper presented aglobal perspective on the principles and applications of space-time-modulated (STM) media. It is shown that magnet-free,wide-bandwidth, linear, and highly versatile microwave nonre-ciprocal systems may be realized by leveraging unique featuresof an STM medium. ST modulation represents a candidate forthe realization of high efficiency components for 5G frequencybands, particularly 3.5 GHz and 26/28 GHz. In addition,STM media can be implemented in conjunction with advancedspace-time multiple-antenna systems [65], which are promis-ing candidates for enhanced-efficiency 4G, 5G and forthcom-ing generations of wireless communication systems, and maybe adopted with link-level ST modulation schemes [66] forthe realization of versatile communication systems. Table Isummarizes the advantages and implementation complexitiesof the STM microwave components presented in this paper.


Fig. 18. STM pure frequency mixer. (a) Generic representation of the spatially aperiodic STM mixer yielding pure frequency conversion from (β0, ω0) to(β1, ω1) [24]. (b) Optimization procedure for mixer synthesis. (c) and (d) Real and imaginary parts of the wavenumber, where aperiodic photonic band gapsprohibit harmonic transitions to undesirable ST harmonics [24].


varactors

Grounding plane

Grounding plane

L

var,KCCvar,2CC2x

z

Matchedload

z∆

L

var,1C1C

2L1 K

K

0 apε ε [1 δ+ m,ap mcos( )z tβ ω− ]

0 apµ µ m m,ap mcos( )V z tβ ω− ( ; )z ω

( ; )z ω

m mcos( )V tω

ε

(a)

signal generator

signal generator

1(input, )ω

mixerDiplexer

DC powersupply

signal analyzer

DC block

Bias-T

Space-time modulated

p(pump wave, )ω

Agilent E8257D

Agilent E8267D Agilent E4440A

(b)

0 0.5 1 1.5 2 2.5−80

−70

−60

−50

−40

−30

−20

−10

0

2(

)(d

Bm

)L

E

9/(2 ) 10ω π ×

ωm

2 ωm

3 ωm

0ω

mω

0ω−

0ω 2

0ω 3

ωm

0ω+

2

ωm

0ω+

3

ωm

0ω−

3

ωm

0ω−

21

ωω

m0

ω+

=

Periodic mod.Aperiodic mod.

ωm

0ω+

2

(c)

Fig. 19. STM pure frequency mixer. (a) Experimental implementation using an array of STM varactors distributed on top of an aperiodic spatially variantconductor-backed coplanar waveguide [24]. (b) Experimental set-up. (c) Experimental comparison of the output spectra of the periodic and the dispersion-tailored aperiodic STM structures, wherein the dispersion-tailored aperiodic STM mixer exhibits negligible power levels of the unwanted harmonics [24].

TABLE IADVANTAGES AND DISADVANTAGES OF THE STM MICROWAVE COMPONENTS PRESENTED IN SEC. IV

STMcomponents

ISOLATORS:i) Sec. IV-A1 [11];ii) Sec. IV-A2 [45];iii) Sec. IV-A3 [18]

CIRCULATOR:Sec. IV-B [21]

FRONT-ENDS:i) Sec. IV-C1 [43];ii) Sec. IV-C2 [30];iii) Sec. IV-C3 [34]

PURE FREQ.MIXER:Sec. IV-D [24]

Sizei) Moderate;ii) Large;iii) Small

Smalli) Moderate,ii) Small,iii) Small

Moderate

Implementationcomplexities

Variable capacitors (orinductors) are required

Variable capaci-tors (or induc-tors) are required

Variable capacitors(or inductors) arerequired

Variable capaci-tors (or induc-tors) are required

Frequency band-width

i) Fair;ii) Narrow;iii) Ultra-wide

Narrowi) Broadband;ii) Narrow;iii) Narrow

Narrow

Tunable Yes Yes Yes Yes

External RF biasrequired Yes, Yes, No Yes Yes Yes

Integrablewith circuittechnology

Yes Yes Yes Yes

Advantages overconventionaltechniques

magnetless, linear,lightweight, ([18]:Self-biased andbroadband)

Magnetless, lin-ear, lightweight

Magnetless, linearand lightweight

No undesiredmixing products,power efficient


REFERENCES

[1] A. L. Cullen, “A travelling-wave parametric amplifier,” Nature, vol. 181,p. 332, Feb. 1958.

[2] P. K. Tien, “Parametric amplification and frequency mixing in propa-gating circuits,” J. Appl. Phys., vol. 29, no. 9, pp. 1347–1357, Sept.1958.

[3] A. Cullen, “Theory of the travelling-wave parametric amplifier,” Pro-ceedings of the IEE-Part B: Electronic and Communication Engineering,vol. 107, no. 32, pp. 101–107, 1960.

[4] E. S. Cassedy and A. A. Oliner, “Dispersion relations in time-spaceperiodic media: part I-stable interactions,” Proc. IEEE, vol. 51, no. 10,pp. 1342 – 1359, Jun. 1963.

[5] S. T. Peng, E. S. Cassedy, and B. R. Rao, “The sonic region forpropagation in a parametric medium with harmonic pump modulation,”Proc. IEEE, vol. 57, no. 2, pp. 224–225, Feb. 1969.

[6] R. Chu and T. Tamir, “Guided-wave theory of light diffraction byacoustic microwaves,” IEEE Trans. Microw. Theory Techn., vol. 17,no. 11, pp. 1002–1020, Nov. 1969.

[7] J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interbandtransitions in photonic crystals,” Phys. Rev. B, vol. 59, pp. 1551–1554,Jan. 1999.

[8] P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson,“Inducing photonic transitions between discrete modes in a siliconoptical microcavity,” Phys. Rev. Lett., vol. 100, no. 3, p. 033904, 2008.

[9] Z. Yu and S. Fan, “Complete optical isolation created by indirectinterband photonic transitions,” Nat. Photonics, vol. 3, pp. 91 – 94,Jan. 2009.

[10] H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocityinduced by interband photonic transition on a silicon chip,” Phys. Rev.Lett., vol. 109, p. 033901, Jul. 2012.

[11] S. Taravati, N. Chamanara, and C. Caloz, “Nonreciprocal electromag-netic scattering from a periodically space-time modulated slab andapplication to a quasisonic isolator,” Phys. Rev. B, vol. 96, no. 16, p.165144, Oct. 2017.

[12] S. Taravati and A. A. Kishk, “Introduction to equilibrated space-time-varying electromagnetic systems,” in Photonics North 2018, Montreal,Canada, Jun. 2018.

[13] S. Taravati, “Giant linear nonreciprocity, zero reflection, and zero bandgap in equilibrated space-time-varying media,” Phys. Rev. Appl., vol. 9,no. 6, p. 064012, 2018.

[14] S. Taravati and A. A. Kishk, “Dynamic modulation yields one-way beamsplitting,” Phys. Rev. B, vol. 99, no. 7, p. 075101, Jan. 2019.

[15] K. A. Lurie, D. Onofrei, W. C. Sanguinet, S. L. Weekes, and V. V.Yakovlev, “Energy accumulation in a functionally graded spatial-temporal checkerboard,” IEEE Antennas Wirel. Propagat. Lett., vol. 16,2017.

[16] S. Bhandare, S. K. Ibrahim, D. Sandel, H. Zhang, F. Wust, and R. Noe,“Novel nonmagnetic 30-dB traveling-wave single-sideband optical iso-lator integrated in III/V material,” IEEE J. Sel. Top. Quantum Electron.,vol. 11, no. 2, pp. 417–421, 2005.

[17] H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocityinduced by interband photonic transition on a silicon chip,” Phys. Rev.Lett., vol. 109, no. 3, p. 033901, 2012.

[18] S. Taravati, “Self-biased broadband magnet-free linear isolator based onone-way space-time coherency,” Phys. Rev. B, vol. 96, no. 23, p. 235150,Dec. 2017.

[19] D. C. Serrano, A. Alu, and J. Gomez-Diaz, “Magnetic-free nonreciprocalphotonic platform based on time-modulated graphene capacitors,” Phys.Rev. B, 2018.

[20] S. Qin, Q. Xu, and Y. E. Wang, “Nonreciprocal components withdistributedly modulated capacitors,” IEEE Trans. Microw. Theory Techn.,vol. 62, no. 10, pp. 2260–2272, 2014.

[21] N. A. Estep, D. L. Sounas, and A. Alu, “Magnetless microwave circula-tors based on spatiotemporally modulated rings of coupled resonators,”IEEE Trans. Microw. Theory Techn., vol. 64, no. 2, pp. 502–518, 2016.

[22] N. Reiskarimian, J. Zhou, and H. Krishnaswamy, “A CMOS passiveLPTV nonmagnetic circulator and its application in a full-duplex re-ceiver,” IEEE Journal of Solid-State Circuits, vol. 52, no. 5, pp. 1358–1372, 2017.

[23] N. Reiskarimian, A. Nagulu, T. Dinc, and H. Krishnaswamy, “Inte-grated conductivity-modulation-based RF magnetic-free non-reciprocalcomponents: Recent results and benchmarking,” IEEE Antennas Wirel.Propagat. Lett., 2018.

[24] S. Taravati, “Aperiodic space-time modulation for pure frequency mix-ing,” Phys. Rev. B, vol. 97, no. 11, p. 115131, 2018.

[25] A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfacesand lorentz nonreciprocity,” Opt. Mat. Exp., vol. 5, no. 11, pp. 2459–2467, Oct. 2015.

[26] Y. Hadad, D. Sounas, and A. Alu, “Space-time gradient metasurfaces,”Phys. Rev. B, vol. 92, no. 10, p. 100304, 2015.

[27] Y. Shi and S. Fan, “Dynamic non-reciprocal meta-surfaces with arbitraryphase reconfigurability based on photonic transition in meta-atoms,”Appl. Phys. Lett., vol. 108, no. 2, p. 021110, 2016.

[28] M. M. Salary, S. Jafar-Zanjani, and H. Mosallaei, “Electrically tunableharmonics in time-modulated metasurfaces for wavefront engineering,”New J. Phys., vol. 20, no. 12, p. 123023, 2018.

[29] S. Taravati and C. Caloz, “Space-time modulated nonreciprocal mixing,amplifying and scanning leaky-wave antenna system,” in IEEE AP-S Int.Antennas Propagat. (APS), Vancouver, Canada, 2015.

[30] Y. Hadad, J. C. Soric, and A. Alu, “Breaking temporal symmetries foremission and absorption,” Proc. Natl. Acad. Sci. U.S.A., vol. 113, no. 13,pp. 3471–3475, Mar. 2016.

[31] D. Ramaccia, D. L. Sounas, A. Alu, F. Bilotti, and A. Toscano,“Non-reciprocity in antenna radiation induced by space-time varyingmetamaterial cloaks,” IEEE Antennas Wirel. Propagat. Lett., 2018.

[32] S. Taravati and A. A. Kishk, “Advanced wave engineering via obliquelyilluminated space-time-modulated slab,” IEEE Trans. Antennas Propa-gat., 2018.

[33] A. Perez, “What is sonic boom: Definition and causes,” 2017.[Online]. Available: http://stillunfold.com/science/what-is-sonic-boom-definition-causes.

[34] S. Taravati and C. Caloz, “Mixer-duplexer-antenna leaky-wave systembased on periodic space-time modulation,” IEEE Trans. Antennas Prop-agat., vol. 65, no. 2, pp. 1–11, Feb. 2017.

[35] P. Tien and H. Suhl, “A traveling-wave ferromagnetic amplifier,” Pro-ceedings of the IRE, vol. 46, no. 4, pp. 700–706, 1958.

[36] M. M. Salary, S. Jafar-Zanjani, and H. Mosallaei, “Time-varyingmetamaterials based on graphene-wrapped microwires: Modeling andpotential applications,” Physical Review B, vol. 97, no. 11, p. 115421,2018.

[37] D. K. Kalluri, Electromagnetics of Time Varying Complex Media:Frequency and Polarization Transformer, 2nd ed. CRC, 2010.

[38] V. Turgul, T. Nesimoglu, and B. Yarman, “A study on RF/microwavetunable inductor topologies,” in 13th Mediterranean Microwave Sympo-sium (MMS), Saida, Lebanon, Sept. 2013.

[39] M. Amirpour, S. Akbari, E. A. Sani, and M. N. Azarmanesh, “Varactorbased tunable inductor design,” in Electrical Engineering (ICEE), 201523rd Iranian Conference on. IEEE, 2015, pp. 1419–1422.

[40] Y. Yoshihara, H. Sugawara, H. Ito, K. Okada et al., “Wide tuningrange LC-VCO using variable inductor for reconfigurable RF circuit,”IEICE transactions on fundamentals of electronics, communications andcomputer sciences, vol. 88, no. 2, pp. 507–512, 2005.

[41] M. Vroubel, Y. Zhuang, B. Rejaei, and J. N. Burghartz, “Integratedtunable magnetic RF inductor,” IEEE Electron Device Letters, vol. 25,no. 12, pp. 787–789, 2004.

[42] J. James, J. Boys, and G. Covic, “A variable inductor based tuningmethod for ICPT pickups,” in the 7th International Power EngineeringConference, Singapore, Jun. 2005.

[43] S. Qin, Q. Xu, and Y. E. Wang, “Nonreciprocal components withdistributedly modulated capacitors,” IEEE Trans. Microw. Theory Techn.,vol. 62, no. 10, pp. 2260–2272, Oct. 2014.

[44] J. Manley and H. E. Rowe, “Some general properties of nonlinearelements-part I: General energy relations,” Proc. IRE, vol. 44, no. 7,pp. 904 – 913, Jul. 1956.

[45] N. Chamanara, S. Taravati, Z.-L. Deck-Leger, and C. Caloz, “Op-tical isolation based on space-time engineered asymmetric photonicbandgaps,” Phys. Rev. B, vol. 96, no. 15, p. 155409, Oct. 2017.

[46] Y. Hadad, D. L. Sounas, and A. Alu, “Space-time gradient metasurfaces,”Phys. Rev. B, vol. 92, no. 10, p. 100304, Sept. 2015.

[47] Y. Shi and S. Fan, “Dynamic non-reciprocal meta-surfaces with arbitraryphase reconfigurability based on photonic transition in meta-atoms,”Appl. Phys. Lett., vol. 108, p. 021110, Jan. 2016.

[48] N. A. Estep, D. L. Sounas, J. Soric, and A. Alu, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys., vol. 10, pp. 923–927, Nov. 2014.

[49] S. Taravati, “Application of space-and time-modulated dispersion engi-neered metamaterials to signal processing and magnetless nonreciproc-ity,” Ph.D. dissertation, Ecole Polytechnique de Montreal, 2017.

[50] S. Taravati and M. Khalaj-Amirhosseini, “Generalised single-sectionbroad-band asymmetrical Wilkinson power divider,” IET Microw. An-tennas Propagat., vol. 6, no. 10, pp. 1164 –1171, Jul. 2012.


[51] T. Dhaene, L. Martens, and D. D. Zutter, “Transient simulation of ar-bitary nonuniform inetrconnection structures characterized by scatteringparameters,” IEEE Trans. Circ. Systems, vol. 39, p. 928–937, Nov. 1992.

[52] J. E. Schutt-Aine, “Transient analysis of nonuniform transmission lines,”IEEE Trans. Circ. Systems, vol. 39, p. 378–385, May 1992.

[53] J. D. Schwartz, M. M. Guttman, J. A. na, and D. V. Plant, “Multichannelfilters using chirped bandgap structures in microstrip technology,” IEEEMicrow. Wireless Compon. Lett., vol. 17, no. 8, pp. 577–579, Aug. 2007.

[54] S. Taravati and M. Khalaj-Amirhosseini, “Design method for matchingcircuits of general multiplexers,” IET Microw. Antennas Propagat.,vol. 7, no. 4, pp. 237–244, Mar. 2013.

[55] A. V. Beljaev, A. P. Krenitskiy, V. P. Meschanov, and L. V. Shikova,“Directional filters on coupled nonuniform TEM transmission lines,”IEEE Trans. Microw. Theory Techn., vol. 52, no. 1, pp. 133–138, Jan.2004.

[56] S. Taravati and M. Khalaj-Amirhosseini, “An efficient method of design-ing dual- and wide-band power dividers with arbitrary power division,”Int. J. RF Microw. Comput. Aided Eng., vol. 24, no. 1, p. 118–126, Jan.2013.

[57] A. Lujambio, I. Arnedo, M. Chudzik, I. Arregui, T. Lopetegi, andM. A. G. Laso, “Dispersive delay line with effective transmission-type operation in coupled-line technology,” IEEE Trans. Microw. TheoryTechn., vol. 21, no. 9, pp. 459–461, Sept. 2011.

[58] S. Taravati, S. Gupta, Q. Zhang, and C. Caloz, “Enhanced bandwidthand diversity in real-time analog signal processing (R-ASP) usingnonuniform C-section phasers,” IEEE Microw. Wireless Compon. Lett.,vol. 26, no. 9, pp. 663–665, Sept. 2016.

[59] M. Khalaj-Amirhosseini, “Wideband or multiband complex impedancematching using microstrip nonuniform transmission lines,” Prog. Elec-tromag. Res., vol. 66, p. 15–25, 2006.

[60] S. Taravati, “Realization of compact broadband rat-race coupler,” Jour-nal of Electromagnetic Waves and Applications, vol. 26, no. 13, p.1708–1715, Sept. 2012.

[61] M. Chudzik, I. Arnedo, A. Lujambio, I. Arregui, I. Gardeta, F. Teberio,J. A. na, D. Benito, M. A. G. Laso, and T. Lopetegi, “Design oftransmission-type nth-order differentiators in planar microwave technol-ogy,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 11, pp. 3384–3394, Nov. 2012.

[62] S. Taravati and M. Khalaj-Amirhosseini, “Compact dual-band stublessbranch-line coupler,” Journal of Electromagnetic Waves and Applica-tions, vol. 26, no. 10, pp. 1323–1331, Jul. 2012.

[63] J. D. Schwartz, J. A. na, and D. V. Plant, “A fully electronic system forthe time magnification of ultra-wideband signals,” IEEE Trans. Microw.Theory Techn., vol. 55, no. 2, pp. 327–334, Feb. 2007.

[64] S. Taravati, “Miniaturized wide band rat race coupler,” Int. J. RF Microw.Comput. Aided Eng., vol. 23, no. 6, pp. 675–681, Oct. 2013.

[65] F. Mehran and R. G. Maunder, “Wireless MIMO systems employingjoint turbo-like STBC codes with bit-level algebraically-interleavedURSCs,” in Wireless Symposium (IWS), 2013 IEEE International, Bei-jing, China, Apr. 2013.

[66] K. Nikitopoulos, F. Mehran, and H. Jafarkhani, “Space-time super-modulation: Concept, design rules, and its application to joint mediumaccess and rateless transmission,” IEEE Trans. Wirel. Commun., vol. 16,no. 12, pp. 8275–8288, 2017.

ACCEPTED AND TO APPEAR IN THE IEEE MICROWAVE …

Documents

Transcript of ACCEPTED AND TO APPEAR IN THE IEEE MICROWAVE …