DDynamic Modulation Yields One-Way Beam Splitting
Sajjad Taravati , and Ahmed A. Kishk Department of Electrical and Computer Engineering,Concordia University, Montr´eal,Quebec H3G 2W1, Canada Department of Electrical and Computer Engineering,University of Toronto, Toronto,Ontario M5S 2E4, Canadaemail: [email protected] (Dated: January 23, 2019)This article demonstrates the realization of an extraordinary beam splitter, exhibiting one-waybeam splitting-amplification. Such a dynamic beam splitter operates based on nonreciprocal andsynchronized photonic transitions in obliquely illuminated space-time-modulated (STM) slabs whichimpart the coherent temporal frequency and spatial frequency shifts. As a consequence of such un-usual photonic transitions, a is exhibited by the STM slab. Beam splitting is a vital operation forvarious communication systems, including circuit quantum electrodynamics, and signal-multiplexingand demultiplexhg. Despite the beam splitting is conceptually a simple operation, the performancecharacteristics of beam splitters significantly influence the repeatability and accuracy of the entiresystem. As of today, there has been no approach exhibiting a nonreciprocal beam splitting accompa-nied with transmission gain and an arbitrary splitting angle. Here, we show that oblique illuminationof a periodic and semi-coherent dynamically-modulated slab results in coherent photonic transitionsbetween the incident light beam and its counterpart space-time harmonic (STH). Such transitionsintroduce a unidirectional synchronization and momentum exchange between two STHs with sametemporal frequencies, but opposite spatial frequencies. Such a beam splitting technique offers highisolation, transmission gain and zero beam tilting, and is expected to drastically decrease the re-source and isolation requirements in communication systems. In addition to the analytical solution,we provide a closed-form solution for the electromagnetic fields in STM structures, and accordingly,investigate the properties of the wave isolation and amplification in subluminal, superluminal andluminal ST modulations.
I. INTRODUCTION
Beam splitters are quintessential elements of commu-nication systems [1–7]. In the microwave regime, beamsplitters are required for the generation of single pho-tons in the circuit quantum electrodynamics [1, 8–13],heterodyne mixer arrays [3], and wave engineering andsignal-multiplexing and demultiplexhg [4–7]. However,the realization of microwave on-chip beam splitters is stillunder research and development [1, 2, 14]. In spite of theimmense scientific attempts for the realization of efficientbeam splitters, beam splitters are restricted to reciprocalresponse and suffer from substantial transmission loss.As a consequence, the resource requirements of the over-all system, including demand for high power microwavesources and isolators, will be increased.This paper presents the application of space-time-modulated (STM) structures to extraordinary beamsplitting. As of today, various applications of STM struc-tures have been reported, where normal incidence of thelight beam to the STM structure yields unusual interac-tion with electromagnetic wave [15–21]. These applica-tions include but not limited to the parametric traveling-wave amplifiers [22–25], isolators [18, 26–32], metasur-faces [33–36], pure frequency mixer [37], circulators [38–40], and mixer-duplexer-antenna system [41, 42]. Nev-ertheless, there has been a lack of investigation on theproperties of STM media under oblique incidence and its applications.Here, we introduce a one-way, tunable and highly ef-ficient beam splitter and amplifier based on coherentphotonic transitions through the oblique illumination ofSTM structures. The contributions of this paper are asfollows.1) In contrast to conventional beam splitters which arerestricted to reciprocal response with more than 3 dB in-sertion loss, the proposed STM beam splitter is capable ofproviding nonreciprocal response with transmission gain.It can be also used in antenna applications, where thetransmitted and received waves are engineered appropri-ately.2) We show that the STM beam splitter presentsan efficient performance for both collimated and non-collimated incidence beam with no output beam tilt.This is very interesting as conventional passive beamsplitters suffer from poor performance for non-collimatedbeams and provide an undesired output beam tilt.3) It is demonstrated that the angle of transmissionand the amplitude of the transmitted beams depend onthe ST modulation parameters. Hence, the ST modu-lation parameters provide the leverage for achieving thedesired angle of transmission for the two output beamsof the STM beam splitter. In addition, unequal powerdivision between the output beams can be achieved byvarying the ST modulation parameters.4) Here, we present the first application of obliquely a r X i v : . [ phy s i c s . op ti c s ] J a n illuminated STM slabs. Consequently, for the first time,the scheme and results for the finite difference time-domain (FDTD) simulation results for oblique incidenceto a STM slab at microwave frequencies is presented.5) A closed-form solution is presented that provides adeep insight into the wave propagation inside the STMbeam splitters and the difference between the subluminal,luminal and superluminal ST modulations.6) The analysis of the STM beam splitter is further ac-complished by investigation of its analytical three dimen-sional dispersion diagrams, achieved by Bloch-Floquetdecomposition of space-time harmonics (STHs).Accordingly, the rest of the paper is structured asfollows. Section II presents the operation principle ofthe proposed STM beam splitter. In Sec. III, we de-rive the analytical solution for oblique electromagneticwave propagation inside the STM beam splitter based onthe Bloch-Floquet representation of the electromagneticfields. Then, Sec. IV presents the time and frequencydomains numerical simulation results for the beam split-ting and amplification in the STM beam splitter. Next,the closed form solution will be provided in Sec. V, whichgives a leverage for understanding the wave propagationand transitions in STM structures. A short discussion onpractical realization of superluminal STM structures atdifferent frequencies will be presented in Sec. VI. Finally,Sec. VII concludes the paper. II. OPERATION PRINCIPLE
Figure 1 sketches the nonreciprocal beam transmissionand splitting in a STM slab. By appropriate design ofthe band structure, that is, the ST modulation formatand its associated temporal and spatial modulation fre-quencies, unidirectional energy and momentum exchangebetween the incident wave-under angle of incidence andtransmission θ I = θ T , = 45 ◦ and temporal frequency ω - to the first lower STH-under angle of transmission θ T , − = − ◦ and temporal frequency ω - will occur.Assuming TM y or E y polarization, the electric field ofthe incident light beam in the forward + z -direction maybe expressed as E FI ( x, z, t ) = E e − i [ k x x + k z z − ω t ] , (1)is traveling in the + z -direction under the angle of inci-dence θ I = 45 ◦ and impinges to the periodic STM slab.The x - and z -components of the spatial frequency read k x = k sin( θ I ) and k z = k cos( θ I ), respectively, in which k = ω /v b = ω √ (cid:15) r /c , with ω being the temporal fre-quency of the incident wave, v b denoting the phase veloc-ity in the background medium, (cid:15) r representing the rela-tive electric permittivity of the background medium, and c denoting the speed of light in vacuum. The STM slabassumes a sinusoidal ST-varying permittivity, as (cid:15) ( z, t ) = (cid:15) av + δ (cid:15) sin( qz − Ω t ) , (2) d FT,-1 E FT,0 E BT E BI E FI E , ω , ω a v s i n ε ε + δ z T,0 θ T,-1 θ , ω , ω , ω ( ) q z t − Ω xzy I θ T θ ° = -45 ° = I θ ° = I θ ° = FIG. 1. Schematic of nonreciprocal beam splitting in a STMslab. The slab varies in time two times faster than the inputwave. where (cid:15) av = (cid:15) r + δ (cid:15) is the average permittivity of theslab, δ (cid:15) denotes the modulation strength, Ω = 2 ω is themodulation temporal frequency, and q = 2 k γ , (3)represents the spatial frequency of the modulation, with γ = v m /v b being the ST velocity ratio, where v m and v b are, the phase velocity of the modulation and the back-ground medium, respectively. Since the slab permittivityis periodic in space and time, with spatial frequency q andtemporal frequency 2 ω , the electric field inside the slabmay be decomposed into ST Bloch-Floquet waves as E S ( x, z, t ) = ˆy M (cid:88) m = − M A m e − i ( k x x + k z,m z − ω m t ) , (4a)and H S ( x, z, t ) = 1 η S (cid:104) ˆk S × E S ( x, z, t ) (cid:105) = M (cid:88) m = − M (cid:20) − ˆx k z,m µ ω m + ˆz sin( θ I ) η S (cid:21) A m e − i ( k x x + k z,m z − ω m t ) . (4b)where M → ∞ is the number of STHs. In Eq. (4), η S = (cid:112) µ / ( (cid:15) (cid:15) r ), and A m represents the unknown amplitudeof the m th STH, characterized by the spatial frequency k z,m = β + mq, (4c)and the temporal frequency ω m = ω + m Ω = (1 + 2 m ) ω , (4d)with β being the unknown spatial frequency of the fun-damental harmonic. The unknowns of the electric field,that is, A m and β , will be found through satisfyingMaxwell’s equations.The transmission angle of the m th transmitted STH, θ T ,m , satisfies the Helmholtz relation as k sin ( θ I ) + k m cos ( θ T ,m ) = k m , (5)where k m = ω m /v b denotes the wavenumber of the m thtransmitted STH outside the STM slab. Solving Eq. (5)for θ T ,m yields θ T ,m = sin − (cid:18) k x k m (cid:19) = sin − (cid:18) sin( θ I )1 + m Ω /ω (cid:19) = sin − (cid:18) sin( θ I )1 + 2 m (cid:19) . (6a)Equation (6a) demonstrates the spectral decomposi-tion of the transmitted wave. Consequently, the funda-mental STH, m = 0, and the first lower STH, m = − ω , will be respectivelytransmitted under the angles of transmission of θ T , = θ I = 45 ◦ ,θ T , − = − θ I = − ◦ . (6b)so that they are transmitted under 90 ◦ angle difference,presenting the desired beam splitting. The scatteringangle of the m th STH inside the STM slab reads θ S ,m = tan − (cid:18) k x k z,m (cid:19) . (7)In addition, the transmitted electric field from the slabmay be found as E T ( x, z, t ) = E S ( x, z, t ) e − ik z,m z = ˆy M (cid:88) m = − M A m e − i ( k x x + k z,m [ d + z ] − ω m t ) . (8)The sourceless wave equation reads ∇ E S ( x, z, t ) − c ∂ [ (cid:15) ( z, t ) E S ( x, z, t )] ∂t = 0 . (9)Substituting Eqs. (S1) and (4) into Maxwell’s equations,yields a matrix equation as[ K ] (cid:126)A = 0 , (10a)where [ K ] is a (2 M + 1) × (2 M + 1) matrix with elements K m,m = (cid:15) av − k x + k z,m k ,K m,m − = i δ (cid:15) ,K m,m +1 = − i δ (cid:15) , (10b) FIG. 2. Qualitative representation of the periodic three di-mensional dispersion diagram for the periodic STM slab inFig. 1. The medium is under oblique illumination of θ I = 45 ◦ at the fundamental harmonic m = 0, corresponding to thetemporal frequency ω , where k x, = ˆx k x = ˆx k sin( θ I ). Thelower STH, m = −
1, provides the same temporal frequency asthe fundamental harmonic, | ω m | = | ω (1 + 2 m ) | m = − = ω ,but opposite x -component of the spatial frequency, that is, k x, − = − ˆx k x = − ˆx k sin( θ I ). and where (cid:126)A represents a (2 M + 1) × A m coefficients. Equation (10a) has nontrivial solution ifdet { [ K ] } = 0 . (11)Equation (11) represents the dispersion relation of theSTM beam splitter which provides the unknown spatialfrequency of the fundamental ST harmonic for a givenfrequency, i.e., β ( ω ). After finding the β ( ω ), the[ K ] matrix in Eq. (10a) is known and therefor, the un-known amplitude of the STHs A m will be calculated us-ing Eq. (10a). III. ANALYTICAL 3D DISPERSION DIAGRAM
Figure 2 presents a qualitative illustration of the threedimensional dispersion diagram in the STM medium inFig. 1 achieved using Eq. (11). This diagram is formedby 2 M + 1 periodic set of double cones (here, only m = 0and m = − k x = 0, k z = − mq and ω = − mω , and the slope of v m with respect to k z − k x plane. Consider oblique incidence of a wave, represent-ing the fundamental harmonic m = 0 with temporal fre-quency ω , propagating along [+ x ,+ z ] direction. It ischaracterized by x - and z -components of the spatial fre-quency, k x = ˆx k x and k F z = ˆz k z . The wave impinges tothe medium under the angle of incidence θ I = 45 ◦ and ex-cites an infinite number of (we truncate it to 2 M +1) STHwaves, with different spatial and temporal frequencies of[ k x , k z,m ] and ω m . However, interestingly, the first lowerSTH m = − ω and identical z -component of the spatial fre-quency of k F z, − = k F z, , but opposite x -component of thespatial frequency of k x, − = − k x, . Hence, m = − − x ,+ z ] direction. In general the x -component of the m th STH reads k x,m = − k x, − m − ).Moreover, since ω m = ω − m − , the undesired STHs ac-quire temporal frequency of 2 mω , and far away fromthe fundamental harmonic. Thus, most of the incidentenergy is residing in m = 0 and m = − ω , respectively transmitted under θ T,0 = θ I and θ T,-1 = − θ I transmission angles with 2 θ I angle difference.The exchange of the energy and momentum betweenthe fundamental and first lower harmonic occurs only forthe forward, + z , wave incidence. This may be observedfrom Fig. 2, as the forward harmonics (red circles, where ∂ω/∂k z >
0) are very close, whereas the backward har-monics (grey circles, where ∂ω/∂k z <
0) are far apartfrom each other. Therefore, a nonreciprocal transition ofenergy is achieved from the incident wave under θ I = 45 ◦ to the first STH under θ T,-1 = − ◦ , through the STmodulation under θ mod = 0 ◦ .Figure 3(a) shows the analytical solution for three di-mensional dispersion diagram of the STM medium inFig. 1, computed using Eq. (11) for γ = 1 .
2. For a givenfrequency, this three dimensional diagram provides thetwo dimensional k z /q − k x /q isofrequency diagram of themedium. Figure 3(b) plots the isofrequency diagram at ω/ ω = 0 . ω = ω ), containing an infinite periodicset of circles centered at ( k z /q, k x /q ) = ( − m,
0) with ra-dius γ (0 . m ).It may be seen from Figs. 3(a) and 3(b) that at ω = ω ,the m = 0 and m = − γ > β ± = k ± z,m +1 − k ± z,m [18]. Particularly, as γ in-creases, ∆ β − increases and ∆ β + decreases. As a re-sult, at the limit of γ = 1 the forward harmonics acquiredistances ∆ β + /q = 0, and the backward harmonics ac-quire distances ∆ β − /q = 2. Hence, increasing γ resultsin the significant enhancement in the nonreciprocity ofthe medium, so that the forward harmonic waves tendto merge together (∆ β + →
0) and exchange their en-ergy and momentum, whereas the backward harmonicstend to separate from each other (∆ β + →
2) (Fig. 3(a)).Hence, such a dynamic modulation has nearly no effecton the backward incident beam. - - - . - . - - / q / ω ω / q xk Forward harmonicsBackward harmonics0 m = m = m = m = -3 m = -2 m = -1 m = I θ ° = β + ∆ β − ∆ z k (a) -3 -2 -1 0 1 2-2-1.5-1-0.500.511.52 3 / q z , k z , - k / x k q m = -1 m = -2 m = m = I θ ° = line x ,0 k k + -1 k + F Planeof symmetry m = z k F x ,-1 k (b) FIG. 3. Analytical dispersion diagram of the periodic STMslab in Fig. 1 for γ = 1 . θ I = 45 ◦ corresponding to k x /q =0 . m = − m = 0 and m = − . (a) Three dimen-sional dispersion diagram constituted of an array of periodiccones [18]. (b) Isofrequency diagram at ω = ω presents aninfinite set of circles centered at ( k z /q, k x /q ) = ( − m,
0) withradius γ (0 . m ). IV. NUMERICAL SIMULATION RESULTS
We next verify the above theory by finite differencetime-domain (FDTD) numerical simulation of the dy-namic process through solving Maxwell’s equations. Fig-ure 4 plots the implemented finite-difference time-domainscheme for numerical simulation of the oblique wave im-pinging to the STM beam splitter. We first discretizethe medium to K + 1 spatial samples and M + 1 tempo-ral samples, with the steps of ∆ z and ∆ t , respectively.Next, the finite-difference discretized form of the first two zzz z ∆ z ( 1) K z + ∆ j = j = j = j K = + t = t t = ∆ time space ( z )/ 2 t t = ∆ i = i = i = E y x H z i = M + ( 1) t M t = + ∆ x H x H x HE y E y E y z / 2 t t = ∆ i = z H z H z HE y E y E y E y E y E y E y E y FIG. 4. General representation of the finite-difference time-domain scheme for numerical simulation of the oblique inci-dence of an E y wave to STM beam splitter. Maxwell’s equations for the electric and magnetic fields(considering Eq. (4)) will be simplified to H x | i +1 / j +1 / = (1 − ∆ t ) H x | i − / j +1 / + ∆ tµ ∆ z (cid:0) E y | ij +1 − E y | ij (cid:1) (12a) H z | i +1 / j +1 / = (1 − ∆ t ) H z | i − / j +1 / − ∆ tµ ∆ z (cid:0) E y | ij +1 − E y | ij (cid:1) (12b) E y | i +1 j = (cid:32) − ∆ t(cid:15) (cid:48) | ij (cid:15) | i +1 / j (cid:33) E y | ij + ∆ t/ ∆ z(cid:15) | i +1 / j . (cid:104)(cid:16) H x | i +1 / j +1 / − H x | i +1 / j − / (cid:17) − (cid:16) H z | i +1 / j +1 / − H z | i +1 / j − / (cid:17)(cid:105) (12c)where (cid:15) (cid:48) = ∂(cid:15) ( z, t ) /∂t = − Ω δ (cid:15) cos( qz − Ω t ).Figure 5 shows the numerical simulation results forthe forward oblique wave incidence to the slab, shown xzy I θ T θ FIG. 5. Nonreciprocal beam splitting in periodically STMslab. FDTD numerical simulation for the forward wave inci-dence to the slab, from the left, with θ I = 45 ◦ . in Fig. 1, with (cid:15) r = 1, δ (cid:15) = 0 . γ = 1 . d = 3 λ =3 × π/k , θ I = 45 ◦ and ω = 3 GHz. It may be seenfrom this figure that an efficient beam splitting with sig-nificant transmission gain is achieved in the forward di-rection. Figures 6(a) and 6(b) provide the results forthe oblique wave incidence from the right side and top,respectively, corresponding to θ I = 45 ◦ and θ I = − ◦ .The presented analytical and numerical results demon-strate that the dynamic beam splitter provides a perfectnonreciprocal beam splitting, in the lack of beam tilting.Moreover, it may be seen that, in contrast to conventionalpassive beam splitters, the beam splitting is achieved fora non-collimated beam. Other interesting features maybe presented by changing the modulation parameters ( γ , θ I and (cid:15) av ), including tunable transmission angles, un-equal splitting ratio and unequal angles of transmission.Figure 7 compares the analytical and numerical resultsfor the spectrum of the incident and transmitted electricfields in Fig. 5. This figure shows that 3dB transmissiongain is achieved for each of transmitted beams in the for-ward excitation. Moreover, it may be seen from Fig. 7that the undesired higher order harmonics, at ω = 2 mω ,are sufficiently weak so that the beam splitter safely op-erates at single frequency ω . V. CLOSED FORM SOLUTION FORELECTROMAGNETIC FIELDS
It is shown in Sec. IV that by proper design of theband structure, a pure unidirectional beam splitting canbe achieved in a obliquely illuminated STM slab. The (a)(b)
FIG. 6. Nonreciprocal beam splitting in periodically STMslab. FDTD numerical simulation for the wave incidence tothe slab, (a) From the right with θ I = 45 ◦ . (b) From the top,i.e., θ I = − ◦ . analytical solution of the electromagnetic fields based onthe double Bloch-Floquet decomposition of electromag-netic fields, presented in Sec. III, provides an accuratesolution for the fields scattered by such a slab, which isvery useful. However, such an analytical solution doesnot provide a deep insight into the wave propagation in-side the slab. In particular, it is of great interest to havean intuitive explanation about the effect of different pa-rameters, e.g. δ (cid:15) , γ , k x and k z , on the wave propagationand energy exchange between the incident field m = 0 − − −− ( d B W ) E Analytical-TransmissonAnalytical-TransmissonFDTD-TransmissonFDTD-TransmissonIncident o at 45 o at -45at -45 oo at 45 ω ω ω ω ω o at 45 FIG. 7. Comparison of the analytical and numerical resultsfor the frequency spectrum of the incident and transmittedelectric fields in Fig. 5, i.e., wave incidence to the slab fromthe left with θ I = 45 ◦ . and the excited first lower harmonic m = −
1. Moreover,the accurate analytical solution, based on the mathemat-ical modeling presented in Sec. III, is achieved througha substantial computational cost. To resolve this issue,here we provide an approximate closed form solution forthe electromagnetic fields propagating inside and trans-mitted from the STM beam splitter, which provides aclear picture of the transition between the incident andthe first lower STHs.As we showed in the previous section, given the weaktransition of energy and momentum from the fundamen-tal STH m = 0 to higher order STHS except m = − m = 0 and m = −
1. The electric field is thendefined by E S ( x, z, t ) = a ( z ) e − i ( k x x + k z z − ω t ) + a − ( z ) e i ( − k x x +( q − k z ) z − ω t ) , (13)where a ( z ) and a − ( z ) are the unknown field coeffi-cients. We shall stress that, here the field coefficientsare z -dependent since they include both the amplitudeand the change in the spatial frequency (wavenumber)introduced by the ST modulation. Following the proce-dure provided in the supplemental material in Ref. [43],we insert the electric fields in (13) into the wave equa-tions in (S4), and achieve a coupled differential equationfor the field coefficients, i.e., ddz (cid:20) a ( z ) a − ( z ) (cid:21) = (cid:20) M C C − M − (cid:21) (cid:20) a ( z ) a − ( z ) (cid:21) , (14a)where M = ik k z ( (cid:15) av − (cid:15) r ) ,M − = ik k z − q ) (cid:20) (cid:15) av − (cid:15) r k x + ( q − k z ) k (cid:21) ,C = i δk k z ,C − = i δk k z − q ) . (14b)The solution to the coupled differential equationin (14a) is given by [43] a ( z ) = E ∆ (cid:18) ( M − M − + ∆ ) e M M − ∆ z (15a) − ( M − M − − ∆ ) e M M − − ∆ z (cid:19) ,a − ( z ) = E C − ∆ (cid:16) e M M − ∆ z − e M M − − ∆ z (cid:17) , (15b)where ∆ = (cid:112) ( M − M − ) + 4 C C − . For a given STmodulation ratio γ , the field coefficients in Eq. (15) ac-quire different forms. In general, ST modulation is clas-sified into three categories, i.e., subluminal (0 < γ < v m < v b ), luminal ( γ → v m → v b ), and superluminal( γ > v m > v b ). A. Subluminal and Superluminal ST Modulations
Considering (cid:15) av = (cid:15) r , the a ( z ) and a − ( z ) in Eq. (15)would be a periodic sinusoidal function with respect to z , if ∆ = (cid:112) ( M − M − ) + 4 C C − is imaginary, i.e.,( M − M − ) + 4 C C − <
0. By solving this, we achievean interval for the luminal ST modulation, that is, γ sub < √ (cid:15) av + δ (cid:15) ≤ γ lum ≤ √ (cid:15) av − δ (cid:15) < γ sup , (16)where γ sub , γ lum and γ sup are ST velocity ratio forsubluminal, luminal and superluminal ST modulations,respectively. The interval for luminal ST modulation iscalled sonic regime in analogy with sonic boom effect inacoustics, where an airplane travels with the same speedor faster than the speed of sound. It should be notedthat the luminal ST modulation interval in Eq. (16)is exactly same as the one achieved from the exactanalytical solution [17, 18, 25].Figure 8(a) plots the closed form and FDTD numericalsimulation results for the absolute electric field coefficientinside the slab, with the wave incidence from the left side(forward incidence), considering superluminal ST modu-lation of γ = 1 . δ (cid:15) = 0 .
28. It is seen from thisfigure that both a ( z ) and a − ( z ) possess periodic sinu-soidal form and exhibit a substantial transmission gain at z = 3 λ . Such a transmission gain may be tuned throughthe variation of γ and δ (cid:15) . This result is consistent withthe transmission gain achieved in the FDTD numericalsimulation results in Figs. 5 and 7. The coherence length l c , where both a ( z ) and a − ( z ) acquire their maximumamplitude is found as [43] l c = π (cid:32)(cid:20) k [ (cid:15) av − (cid:15) r ] /k z − q ( γ − / ( γ − (cid:21) + δ k k z ( k z − q ) (cid:33) − . (17)Figure 8(b) plots the result for the superluminal STMslab in Fig. 8(a), except for wave incidence from the rightside (backward incidence). It may be seen from this fig-ure that, in contrast to the forward wave incidence wherea substantial exchange of the energy and momentum be-tween the m = 0 and m = − m = − B. Luminal ST Modulation
It may be shown that the for the luminal ST modula-tion, where γ →
1, the field coefficients in Eq. (15), a ( z )and a − ( z ), acquire pure real (or complex) forms. Thisyields exponential growth of the electric field amplitudealong the STM slab. Hence, considering γ = 1, the totalelectric field inside the STM slab reads E S ( x, z, t ) | γ =1 = E cosh (cid:18) δk k z z (cid:19) e − i ( k x x + k z z − ω t ) (18) − i δk k z E sinh (cid:18) δk k z z (cid:19) e i ( − k x x +( q − k z ) z − ω t ) . Figure 9(a) plots the closed form and FDTD numeri-cal simulation results for the absolute value of the elec-tric field coefficients a ( z ) and a − ( z ) inside the luminal( γ = 1 and δ (cid:15) = 0 .
28) STM slab for forward wave inci-dence. It may be seen from this figure that both a ( z )and a − ( z ) possess a non-periodic exponentially grow-ing profile and exhibit a substantial transmission gainat z ≥ λ . It should be noted that, the solution forthe field coefficients presented in Eqs. (15) and (S21) arevery useful and provide a deep insight into the wave prop-agation inside the STM slab, especially for the luminalST modulation (sonic regime), where the Bloch-Floquet-based analytical solution does not exist since the solutiondoes not converge [17, 18, 25].Figure 9(b) plots the result for the luminal STM slab inFig. 9(a), except for wave incidence from the right side(backward incidence). It may be seen from this figurethat, in contrast to the forward wave incidence, here the Forward incidence | a ( z ) | / 3 z λ ( ) a z ( ) a z − ( ) a z ( ) a z − ( ) a z − ( ) a z -Closed form-Closed form-FDTD simulation-FDTD simulation T r a n s m i ss i on g a i n (a) BAckward incidence | a ( z ) | / 3 z λ ( ) a z ( ) a z − (b) FIG. 8. Closed-form solution results and the FDTD numer-ical simulation results for the z -dependent absolute field co-efficients in Eq. (13), i.e., a ( z ) and a − ( z ), inside the su-perluminal STM beam splitter, with γ = 1 . δ (cid:15) = 0 . incident wave passes through the slab with negligible al-teration and minor transition of energy and momentumto the m = − VI. DISCUSSION ON PRACTICALREALIZATION OF SUPERLUMINAL STMODULATION
To practically realize superluminal ST modulation(here γ = 1 . Forward incidence | a ( z ) | / 3 z λ ( ) a z ( ) a z − ( ) a z ( ) a z − ( ) a z − ( ) a z -Closed form-Closed form-FDTD simulation-FDTD simulation T r a n s m i ss i on g a i n (a) BAckward incidence | a ( z ) | / 3 z λ ( ) a z ( ) a z − (b) FIG. 9. Closed-form solution results and the FDTD numeri-cal simulation results for the z -dependent absolute field coef-ficients in Eq. (13), i.e., a ( z ) and a − ( z ), inside the luminalSTM beam splitter, with γ = 1 and δ (cid:15) = 0 .
28. (a) For-ward wave incidence, where the wave propagates from left toright. (b) Backward wave incidence, where the wave propa-gates from right to left. velocity of light in vacuum [15]. Considering a glass asthe background medium with permittivity > .
5, achiev-ing γ = 1 . • At low frequencies, one may use filter constantswhich are appropriately selected for two weaklycoupled transmission lines [18, 30, 42, 44], one forthe pump and one for the main incident wave. • At ultra-high frequencies, a serpentine transmis-sion line msupports the propagation of the mainincident wave [45], which lowers the phase velocityrelative to the modulation velocity. • At microwave frequencies the pump wave is sup-ported in a closed waveguide [46], thereby achievinga fast phase velocity.In addition, recently, there has been an experimentaldemonstration of time-modulated structure [47], wherethe medium is periodically modulated in time only, rep-resenting the limiting case of an infinite modulation ve-locity, i.e., q = 0 and hence v m = Ω /q → ∞ . VII. CONCLUSION
We have introduced a unidirectional beam splitter andamplifier based on asymmetric coherent photonic tran-sitions in obliquely illuminated space-time-modulated(STM) media. The operation of this dynamic beam split-ter is demonstrated by both the analytical, closed-formand numerical simulation results. While the normally il-luminated STM media have been previously used for therealization of various components, including insulators,parametric amplifiers and nonreciprocal frequency gen-erators, this paper presented the first study investigat-ing the oblique illumination of STM media. Accordingly,this paper proposed a forward-looking application of suchdynamic media. The proposed unidirectional beam split-ter is endowed with unique functionalities, including ad-justable one-way transmission gain, tunable splitting an-gle and arbitrary unequal splitting power ratio, as wellas high isolation, and hence, is expected to substantiallyreduce the source and isolation requirements of commu-nication systems. [1] Christian Schneider,
On-chip superconducting microwavebeam splitter , Ph.D. thesis, Masters thesis, TechnischeUniversit¨at M¨unchen (2014).[2] Jakob Hammer, Sebastian Thomas, Philipp Weber, andPeter Hommelhoff, “Microwave chip-based beam splitterfor low-energy guided electrons,” Phys. Rev. Lett. ,254801 (2015).[3] Boon-Kok Tan and Ghassan Yassin, “A planar beamsplitter for millimeter and submillimeter heterodynemixer array,” IEEE Transactions on Terahertz Scienceand Technology , 664–668 (2017).[4] Ryuichi Watanabe, “A novel polarization-independentbeam splitter,” IEEE Trans. Microw. Theory Techn. ,685–689 (1980).[5] W Zhu, J Shaker, JS Wight, M Cuhaci, and DA McNa-mara, “Microwave spatial beam splitter/combiner usingartificial microwave volume hologram technology,” Elec-tronics Letters , 1 (2006).[6] Ruey-Bing Hwang, Neng-Chieh Hsu, and Cheng-YuanChin, “A spatial beam splitter consisting of a near-zerorefractive index medium,” IEEE Trans. Antennas Prop-agat. , 417–420 (2012).[7] James R Cooper, Sangkil Kim, and Manos MTentzeris, “A novel polarization-independent, free-space,microwave beam splitter utilizing an inkjet-printed, 2-Darray frequency selective surface,” IEEE Antennas Wirel.Propagat. Lett. , 686–688 (2012).[8] Andreas Wallraff, David I Schuster, Alexandre Blais,L Frunzio, R-S Huang, J Majer, S Kumar, Steven MGirvin, and Robert J Schoelkopf, “Strong coupling ofa single photon to a superconducting qubit using circuitquantum electrodynamics,” Nature , 162 (2004).[9] Alexandre Blais, Ren-Shou Huang, Andreas Wallraff,Steven M Girvin, and R Jun Schoelkopf, “Cavity quan-tum electrodynamics for superconducting electrical cir-cuits: An architecture for quantum computation,” Phys. Rev. A , 062320 (2004).[10] Matteo Mariantoni, Frank Deppe, Achim Marx, RudolfGross, Frank K Wilhelm, and Enrique Solano, “Two-resonator circuit quantum electrodynamics: A super-conducting quantum switch,” Phys. Rev. B , 104508(2008).[11] AA Houck, DI Schuster, JM Gambetta, JA Schreier,BR Johnson, JM Chow, L Frunzio, J Majer, MH De-voret, SM Girvin, et al. , “Generating single microwavephotons in a circuit,” Nature , 328 (2007).[12] D Bozyigit, C Lang, L Steffen, JM Fink, C Eichler,M Baur, R Bianchetti, PJ Leek, S Filipp, A Wall-raff, et al. , “Correlation measurements of individual mi-crowave photons emitted from a symmetric cavity,” in Journal of Physics: Conference Series , Vol. 264 (IOPPublishing, 2011) p. 012024.[13] D Bozyigit, C Lang, L Steffen, JM Fink, C Eichler,M Baur, R Bianchetti, PJ Leek, S Filipp, MP Da Silva, et al. , “Antibunching of microwave-frequency photonsobserved in correlation measurements using linear detec-tors,” Nature Physics , 154 (2011).[14] Matteo Mariantoni, Edwin P Menzel, Frank Deppe,MA Araque Caballero, Alexander Baust, Thomas Niem-czyk, Elisabeth Hoffmann, Enrique Solano, Achim Marx,and Rudolf Gross, “Planck spectroscopy and quantumnoise of microwave beam splitters,” Phys. Rev. Lett. ,133601 (2010).[15] E. S. Cassedy, “Dispersion relations in time-space peri-odic media: part II -unstable interactions,” Proc. IEEE , 1154 – 1168 (1967).[16] Z. Yu and S. Fan, “Complete optical isolation created byindirect interband photonic transitions,” Nat. Photonics , 91 – 94 (2009).[17] Sajjad Taravati, “Giant linear nonreciprocity, zero re-flection, and zero band gap in equilibrated space-time-varying media,” Phys. Rev. Appl. , 064012 (2018). [18] Sajjad Taravati, Nima Chamanara, and ChristopheCaloz, “Nonreciprocal electromagnetic scattering from aperiodically space-time modulated slab and applicationto a quasisonic isolator,” Phys. Rev. B , 165144 (2017).[19] Mohammad Mahdi Salary, Samad Jafar-Zanjani, andHossein Mosallaei, “Time-varying metamaterials basedon graphene-wrapped microwires: Modeling and poten-tial applications,” Phys. Rev. B , 115421 (2018).[20] Mohammad Mahdi Salary, Samad Jafar-Zanjani, andHossein Mosallaei, “Electrically tunable harmonics intime-modulated metasurfaces for wavefront engineering,”New Journal of Physics (2018).[21] Sajjad Taravati and Ahmed A. Kishk, “Advancedwave engineering via obliquely illuminated space-time-modulated slab,” IEEE Trans. Antennas Propagat. ,270–281 (2019).[22] A. L. Cullen, “A travelling-wave parametric amplifier,”Nature , 332 (1958).[23] P. K. Tien, “Parametric amplification and frequency mix-ing in propagating circuits,” J. Appl. Phys. , 1347–1357 (1958).[24] PK Tien and H Suhl, “A traveling-wave ferromagneticamplifier,” Proceedings of the IRE , 700–706 (1958).[25] E. S. Cassedy and A. A. Oliner, “Dispersion relations intime-space periodic media: part I -stable interactions,”Proc. IEEE , 1342 – 1359 (1963).[26] JL Wentz, “A nonreciprocal electrooptic device,” Pro-ceedings of the IEEE , 97–98 (1966).[27] Suhas Bhandare, Selwan K Ibrahim, David Sandel,Hongbin Zhang, F Wust, and Reinhold No´e, “Novelnonmagnetic 30-dB traveling-wave single-sideband opti-cal isolator integrated in III/V material,” IEEE J. Sel.Top. Quantum Electron. , 417–421 (2005).[28] H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electri-cally driven nonreciprocity induced by interband pho-tonic transition on a silicon chip,” Phys. Rev. Lett. ,033901 (2012).[29] Nima Chamanara, Sajjad Taravati, Zo´e-Lise Deck-L´eger,and Christophe Caloz, “Optical isolation based on space-time engineered asymmetric photonic bandgaps,” Phys.Rev. B , 155409 (2017).[30] Sajjad Taravati, “Self-biased broadband magnet-free lin-ear isolator based on one-way space-time coherency,”Phys. Rev. B , 235150 (2017).[31] JunHwan Kim, Seunghwi Kim, and Gaurav Bahl, “Com-plete linear optical isolation at the microscale with ul-tralow loss,” Scientific reports , 1647 (2017).[32] Donggyu B Sohn, Seunghwi Kim, and Gaurav Bahl,“Time-reversal symmetry breaking with acoustic pump-ing of nanophotonic circuits,” Nat. Photon. , 91(2018).[33] Y. Hadad, D. L. Sounas, and A. Al`u, “Space-time gra-dient metasurfaces,” Phys. Rev. B , 100304 (2015).[34] Y. Shi and S. Fan, “Dynamic non-reciprocal meta-surfaces with arbitrary phase reconfigurability based onphotonic transition in meta-atoms,” Appl. Phys. Lett. , 021110 (2016).[35] Yu Shi, Seunghoon Han, and Shanhui Fan, “Optical cir-culation and isolation based on indirect photonic tran-sitions of guided resonance modes,” ACS Photonics ,1639–1645 (2017).[36] Zhanni Wu and Anthony Grbic, “A transparent, time-modulated metasurface,” in (IEEE, 2018) pp. 439–441.[37] Sajjad Taravati, “Aperiodic space-time modulation forpure frequency mixing,” Phys. Rev. B , 115131 (2018).[38] Shihan Qin, Qiang Xu, and Yuanxun Ethan Wang,“Nonreciprocal components with distributedly modu-lated capacitors,” IEEE Trans. Microw. Theory Techn. , 2260–2272 (2014).[39] Nicholas A Estep, Dimitrios L Sounas, Jason Soric, andAndrea Al`u, “Magnetic-free non-reciprocity and isolationbased on parametrically modulated coupled-resonatorloops,” Nat. Phys. , 923–927 (2014).[40] Negar Reiskarimian, Aravind Nagulu, Tolga Dinc,and Harish Krishnaswamy, “Integrated conductivity-modulation-based rf magnetic-free non-reciprocal compo-nents: Recent results and benchmarking,” arXiv preprintarXiv:1805.11662 (2018).[41] S. Taravati and C. Caloz, “Space-time modulated nonre-ciprocal mixing, amplifying and scanning leaky-wave an-tenna system,” in IEEE AP-S Int. Antennas Propagat.(APS) (Vancouver, Canada, 2015).[42] Sajjad Taravati and Christophe Caloz, “Mixer-duplexer-antenna leaky-wave system based on periodic space-timemodulation,” IEEE Trans. Antennas Propagat. , 442– 452 (2017).[43] See Supplemental Material for detailed derivations.[44] S Okwit and EW Sard, “Constant-output-frequency, oc-tave tuning range backward-wave parametric amplifier,”IRE Transactions on Electron Devices , 540–549 (1961).[45] S Okwit, MI Grace, and EW Sard, “UHF backward-waveparametric amplifier,” IRE Transactions on MicrowaveTheory and Techniques , 558–563 (1962).[46] Hsiung Hsu, “Backward traveling-wave parametric am-plifiers,” in Solid-State Circuits Conference. Digest ofTechnical Papers. 1960 IEEE International , Vol. 3(IEEE, 1960) pp. 91–92.[47] Jos´e Roberto Reyes-Ayona and Peter Halevi, “Electro-magnetic wave propagation in an externally modulatedlow-pass transmission line,” IEEE Transactions on Mi-crowave Theory and Techniques , 3449–3459 (2016). SUPPLEMENTAL MATERIAL
The beam splitter is placed between z = 0 and z = d , and represented by the space-time-varying permittivity of (cid:15) ( z, t ) = (cid:15) av + δ (cid:15) sin( qz − Ω t ) , (S1)where Ω = 2 ω and q = 2 k z γ . (S2)The electric field inside the beam splitter is defined based on the superposition of the m = 0 and m = − E S ( x, z, t ) = a ( z ) e − i ( k x x + k z z − ω t ) + a − ( z ) e i ( − k x x +( q − k z ) z − ω t ) , (S3)and the corresponding wave equation reads ∂ E ∂x + ∂ E ∂z = 1 c ∂ [ (cid:15) eq ( t, z ) E ] ∂t . (S4)Inserting the electric field in (S3) into the wave equation in (S4) results in (cid:18) ∂ ∂x + ∂ ∂z (cid:19) (cid:104) a ( z ) e − i ( k x x + k z z − ω t ) + a − ( z ) e i ( − k x x +( q − k z ) z − ω t ) (cid:105) = 1 c ∂ ∂t (cid:18)(cid:20) (cid:15) av + δ e i ( qz − ω t ) + δ e − i ( qz − ω t ) (cid:21) (cid:16) a ( z ) e − i ( k x x + k z z − ω t ) + a − ( z ) e i ( − k x x +( q − k z ) z − ω t ) (cid:17) (cid:19) , (S5)and applying the space and time derivatives, while using a slowly varying envelope approximation, yields (cid:20) ( k x + k z ) a ( z ) − ik z ∂a ( z ) ∂z (cid:21) e − i ( k x x + k z z − ω t ) + (cid:20) ( k x + ( q − k z ) ) a − ( z ) − i ( k z − q ) ∂a − ( z ) ∂z (cid:21) e i ( − k x x +( q − k z ) z − ω t ) = ω c (cid:18)(cid:20) (cid:15) av + δ e i ( qz − ω t ) + 9 δ e − i ( qz − ω t ) (cid:21) a ( z ) e − i ( k x x + k z z − ω t ) + (cid:20) (cid:15) av + 9 δ e i ( qz − ω t ) + δ e − i ( qz − ω t ) (cid:21) a − ( z ) e i ( − k x x +( q − k z ) z − ω t ) (cid:19) , (S6)We then multiply both sides of Eq. (S6) with e i ( k x x + k z z − ω t ) , which gives (cid:20) ( k x + k z ) a ( z ) − ik z ∂a ( z ) ∂z (cid:21) + (cid:20) ( k x + ( q − k z ) ) a − ( z ) − i ( k z − q ) ∂a − ( z ) ∂z (cid:21) e i ( qz − ω t ) = ω c (cid:18)(cid:20) (cid:15) av + δ e i ( qz − ω t ) + 9 δ e − i ( qz − ω t ) (cid:21) a ( z ) + (cid:20) (cid:15) av e i ( qz − ω t ) + 9 δ e i qz − ω t ) + δ (cid:21) a − ( z ) (cid:19) , (S7)and next, applying (cid:82) πω dt to both sides of (S7) yields da ( z ) dz = ik k z (cid:18) [ (cid:15) av − (cid:15) r ] a ( z ) + δ a − ( z ) (cid:19) , (S8)which may be cast as da ( z ) dz = M a ( z ) + C a − ( z ) , (S9)where M = ik k z ( (cid:15) av − (cid:15) r ) , C = iδk k z . (S10)2Following the same procedure, we next multiply both sides of (S7) with e − i ( qz − ω t ) , and applying (cid:82) πω dt in bothsides of the resultant, which reduces to (cid:20) ( k x + ( q − k z ) ) a − ( z ) − i ( k z − q ) ∂a − ( z ) ∂z (cid:21) = ω c (cid:18) δ a ( z ) + (cid:15) av a − ( z ) (cid:19) , (S11)which may be cast as da − ( z ) dz = C − a ( z ) + M − a − ( z ) , (S12)where M − = ik k z − q ) (cid:20) (cid:15) av − (cid:15) r k x + ( q − k z ) k (cid:21) , C − = iδk k z − q ) (S13)Equations (S9) and (S12) form a matrix differential equation. We then look for independent differential equationsfor a ( z ) and a − ( z ), which is expressed by d a ( z ) dz − ( M + M − ) da ( z ) dz + ( M M − − C C − ) a ( z ) = 0 , (S14) d a − ( z ) dz − ( M + M − ) da − ( z ) dz + ( M M − − C C − ) a − ( z ) = 0 . (S15)which are second order differential equations. Using the initial conditions of a (0) = E and a − (0) = 0 gives a ( z ) = E ∆ (cid:16) ( M − M − + ∆ ) e M M − ∆ z − ( M − M − − ∆ ) e M M − − ∆ z (cid:17) , (S16) a − ( z ) = C − E ∆ (cid:16) e M M − ∆ z − e M M − − ∆ z (cid:17) , (S17)where ∆ = (cid:112) ( M + M − ) − M M − − C C − ) = (cid:112) ( M − M − ) + 4 C C − . (S18)Assuming an imaginary result for ∆ (which occurs for subluminal and superluminal space-time modulations), a ( z )and a − ( z ) acquire a periodic sinusoidal form, where the maximum amplitude of them occur at the coherence length z = l c , where ddz a ( z ) | z = l c = ddz a − ( z ) | z = l c = 0 , (S19)which corresponds to l c = π (cid:32)(cid:20) k [ (cid:15) av − (cid:15) r ] /k z − q ( k z − q ) / ( k z − q/ (cid:21) + δ k k z ( k z − q ) (cid:33) − . (S20)For luminal space-time modulation, where γ → q → k z , a ( z ) and a − ( z ) acquire exponentially growingprofile, and hence, the total electric field inside the slab reads E ( x, z, t ) = E cosh (cid:18) δk k z z (cid:19) e − i ( k x x + k z z − ω t ) − i δk k z E sinh (cid:18) δk k z z (cid:19) e i ( − k x x +( q − k z ) z − ω t ) , (S21)which demonstrates the wave amplification of both m = 0 and m = −−