Giant Non-reciprocity Near Exceptional Point Degeneracies
GGiant Non-reciprocity Near Exceptional Point Degeneracies
Roney Thomas ∗ , Huanan Li ∗ , F. M. Ellis, & Tsampikos Kottos Department of Physics, Wesleyan University, Middletown, CT-06459, USA
We show that gyrotropic structures with balanced gain and loss that respect anti-linear symmetriesexhibit a giant non-reciprocity at the so-called exact phase where the eigenfrequencies of the isolatednon-Hermitian set-up are real. The effect occurs in a parameter domain near an exceptional point(EP) degeneracy, where mode-orthogonality collapses. The theoretical predictions are confirmednumerically in the microwave domain, where a non-reciprocal transport above 90dB is demonstrated,and are further verified using lump-circuitry modeling. The analysis allows us to speculate theuniversal nature of the phenomenon for any wave system where EP and gyrotropy can co-exist.
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I. INTRODUCTION
Exceptional points (EP) are non-Hermitian degenera-cies where both eigenvalues and eigenvectors coalesce [1].Originally treated as mathematical curiosities [2–5], thesedegeneracies have been now recognized as a source ofmany counter-intuitive phenomena, some of which canbe exploited for technological purposes. Examples in-clude loss-induced transparency [6], unidirectional invis-ibility [7–9], lasing mode selection [10], lasing revivalsand suppression [11], directional lasing [12], hypersensi-tive sensors [13] etc.The wealth of these results and the demonstrated ca-pability of the researchers to utilize EPs in order to de-sign novel devices, motivated us here to employ them forthe realization of a new class of photonic isolators andcirculators with an extraordinary (giant) non-reciprocaltransport. The proposed structures are linear, they in-volve gyrotropic elements, and they operate in a param-eter domain, near an EP degeneracy, where they are sta-ble i.e. the eigenfrequencies of the associated isolatedset-up are real [14, 15]. The latter two “conflicting” re-quirements can be satisfied simultaneously by a class ofnon-Hermitian systems which involve balanced gain andloss mechanisms and which respect antilinear symmetries[3]. The parameter domain for which the eigenfrequenciesare real (stable domain) is known as exact phase whilethe domain for which the spectrum consists of conjugatepairs of complex eigenvalues (unstable domain) is knownas broken symmetry phase. The transition between thesetwo phases occurs via an EP [3]. A prominent exampleof such antilinear systems are structures with parity-time( PT ) symmetry [6–11, 15–26].In this paper we demonstrate the EP-induced giantnon-reciprocity in the microwave domain and establishits universal nature by evincing it in a seemingly differentframework of lumped electronic circuitry. The frequencyfor which the giant non-reciprocity occurs depends onthe values of the gain and loss parameter and the ap- ∗ The first two authors contributed equally to this work plied magnetic field. Our approach provides several de-grees of reconfigurability, thus constituting an alternativepathway [19, 20, 23–25, 27–32] towards enhancing non-reciprocal wave transport.The structure of the paper is as follows. In the nextsection II we present the photonic structure. Specifi-cally in subsection II.A we analyze the ”evolution” ofthe eigenfrequencies of the isolated set-up as a functionof the gain and loss parameter while in subsection II.B wepresent the numerical results for the scattering propertiesof this structure. In subsection II.C we analyze theoreti-cally using coupled mode theory the transport character-istics of the photonic structure and compare our theoret-ical results for the non-reciprocal transmission with thenumerical data. At section III we analyze numericallya user-friendly model of coupled LRC lump circuits andshow that also this system demonstrate the same strongnon-reciprocal transport. Our conclusions are given atthe section IV. H I / P O / P I / P M i c r o c a v i t y M i c r o c a v i t y H G a i n L o ss G a i n L o ss d ww g W a v e g u i d e Y I G -4 Energy density (J/m ) x 10 -5 (a) (b) w xy z x = FIG. 1: Schematic of the photonic structure: two half-wavemicrostrip resonators are end-coupled to a bus waveguide. Auniformly distributed gain or loss material property augmentsthe region beneath each of the resonators within the YIG-substrate. The substrate is exposed to an external bias field,H , in the y -direction. For an appropriate value of the gainand loss parameter γ the transmission in the forward directiontake values of order of unity (a) while it is essentially zero inthe backward direction (b). a r X i v : . [ phy s i c s . c l a ss - ph ] J a n II. PHOTONIC STRUCTURE
We consider the structure shown in Fig. 1. It con-sists of a parallel pair of half-wave microstrip resonators(dimer) end-coupled to a bus waveguide as schematicallyillustrated in Fig. 1. The microstrip resonator dimer andthe waveguide are situated on top of an 8.75 mm thickferrite substrate with a ground plane on the lower sur-face. The length, l , of each microstrip is 24.5 mm, whichcorresponds to an uncoupled half-wave resonance of ap-proximately 1.24 GHz. The widths w and w of themicrostrips and bus waveguide are set at 3.5 mm and3.0 mm respectively, the latter matching the 56 Ohmsimpedance of the input bus ports. The distance, d , be-tween the two microstrip resonators, is set to 20 mm andthe end-coupled gap g between the microstrip resonatordimer and the bus waveguide is 0.5 mm. All metallic sur-face structures are defined as zero-thickness, perfect elec-tric conductors. A relative dielectric permittivity (cid:15) r = 15is used for the ferrite substrate [33, 34] matching YttriumIron Garnet (YIG). In all our simulations, gain and lossare confined to the spatial domain beneath each of themicrostrip resonators and implemented by introducingan imaginary part of the complex permittivity definedas (cid:15) r = 15(1 ± iγ ), wherein γ denotes the gain and lossparameter. A practical way of implementing loss or am-plification (gain) locally (within the microcavities) canbe achieved electronically via discrete electronic (loss) orgain devices such as a (resistor), transistor, or tunneldiode [35, 36].A static magnetic bias field, H , is applied along the y -direction through the substrate material having ananisotropic magnetic permeability tensor, ˆ µ , given by:ˆ µ = µ µ r iκ r − iκ r µ r ; µ r = 1 + κ r ; κ r = ωω m ω − ω , (1)where ω = µ γ e H , ω m = µ γ e M s . Here, µ and ω corresponds to the permeability of free space and an-gular frequency, ω corresponds to the precession fre-quency of an electron in the applied magnetic field bias, H = 1 . × A/m, ω m is the electron Larmor fre-quency at the saturation magnetization, M s = 1 . × A/m of the ferrite medium, and γ e is the gyromagneticconstant of 1.76 × rad/sT.The whole structure satisfies a combined mirror-timesymmetry with respect to the yz -plane at x = 0. Themirror-symmetry operator M is linear and it is associ-ated with a reflection ( x, y, z ) → ( − x, y, z ) around theorigin. The time reversal operator T is antilinear and itis associated with a complex conjugation together witha simultaneous inversion of the magnetic field vectors, (cid:126)H → − (cid:126)H . The mirror-time reversal symmetry belongsto the class of anti-linear symmetries, part of which isalso the parity-time ( PT ) symmetry. In order to stressthis similarity (x-axis parity and to be in direct contactwith the vast community that studies transport of PT - symmetric systems), we will abbreviate below the mirror-time reversal symmetry with the letters ˜ PT .Below we first analyze the parametric evolution of theeigen-frequencies versus the gain and loss parameter ofthe two microstrip system in the absence of the buswaveguide. We refer to this as the “ isolated ” set-up.Its scattering analogue is constructed by passing the buswaveguide near one end of the micro-strip pair (see Fig.1). We refer to this as the “ scattering ” set-up.The electromagnetic propagation is described by theMaxwell’s equations (cid:126) ∇ × (cid:126)E = i ωc ˆ µ (cid:126)H ; (cid:126) ∇ × (cid:126)H = − i ωc ˆ (cid:15) (cid:126)E (2)where (cid:126)E is the electric (cid:126)H is the magnetic field. Theseequations supplemented by Eqs. (1) together with theappropriate boundaries dictated by our design of Fig. 1describe the wave propagation from the structure. Thelatter is simulated with COMSOL’s 3D-finite elementelectromagnetic (FEEM) numerical software [37]. For ac-curacy of the numerical results, each domain of the struc-ture comprised of fine mesh element sizes of ≈ λ m /13within the substrate region and ≈ λ m /8 for the surround-ing air regime, where λ m is the wavelength inside themedium. A. Isolated set-up
We investigate the MT -symmetry phase transition forthe isolated set-up of Fig. 1 using COMSOL’s eigenfre-quency simulation. When γ = 0, the coupled microstripresonators support two low-order resonant modes, whichhave a symmetric (lower frequency ω s ) and an antisym-metric (higher frequency ω a ) configuration. For γ = 0the associated eigenfrequencies have the same imaginaryvalue I m { ω } = η resulting from weak coupling to theperfectly absorbing ends. As γ increases the real partof the eigenfrequencies of the modes changes (see Fig. 2)while the associated imaginary part remains the same[38]. In this domain ( exact phase ) [3], the associatedeigenmodes respect the ˜ PT symmetry. At a critical valueof the gain and loss parameter γ ˜ PT ≈ .
26, the eigenval-ues and eigenvectors coalesce and the system experiencean EP degeneracy. At the broken phase correspondingto γ > γ ˜ PT the real part of the eigenfrequencies remaindegenerate while the imaginary part bifurcates into twovalues. We refer to this transition as a spontaneous ˜ PT -symmetric phase transition . The value of γ ˜ PT dependson the value (and spatial domain) of the applied magneticfield H . B. Scattering set-up
Next we proceed with the analysis of the transmissionproperties of the scattering set-up of Fig. 1. Forward(FWD), or backward (BWD) propagation of radiation is
FIG. 2: Parametric evolution of the real and the imaginaryparts of the eigen-frequencies vs γ for the isolated set-up ofFig. 1. A uniform magnetic field H is imposed on the sub-strate. At γ = 0 we have a non-zero imaginary part due toleakage from the cavities. At γ = γ ˜ PT ≈ .
26 an EP degen-eracy occurs. defined in the context of the 56 Ohm ports, impedancematched to the transverse electromagnetic (TEM) modesfrom the left and right ends of the bus waveguide shownschematically in Fig. 1. Our analysis will concentrateon γ -values for which the system is in the exact phasei.e. γ ≤ γ ˜ PT . To quantify the dependence of the non-reciprocal effect, we introduce the nonreciprocity param-eter NR (measured in dB), N R ( γ ) = 10 × max ω (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) log T B T F (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) , (3)where T F and T B are the transmittances obtained for theFWD and BWD cases, respectively. Our numerical in-vestigation indicates that the maximum values of N R are achieved in the proximity of the symmetric resonantfrequency ω s . We will therefore focus on this frequencydomain. In Figs. 3a-c we show some typical transmis-sion spectra for γ = 0, 0.1675 and 0.18, respectively. Notethat at ω ≈ ω s the BWD transmittance T B becomes es-sentially zero while T F = O (1). Specifically for γ = 0(see Fig. 3a) a non- reciprocal transmission at ω s canbe as high as 18 dB . A higher degree of non-reciprocity N R = 42 . dB occurs for γ = 0 . γ = 0 .
18 leads to a decrease of non-reciprocity to
N R ≈ . dB (see Fig. 3c).The simulation results for N R ( γ ) and its giant en-hancement at some critical gain and loss value γ NR isreported as the solid circles part of Fig. 4b where weshow the degree of non-reciprocity N R versus γ . Thenon-monotonic behavior of N R , and the associated max-ima, constitute the main result of our study, theoreticallydiscussed in the next section. -50-250
Theory T B Numerics T B Theory T F Numerics T F -50-250 7.85 7.9 7.95 8-50-250 g = 0g = 0.18g = 0.1675 Frequency w (nsec -1 ) T r a n s m itt a n ce ( d B ) (a)(b)(c) FIG. 3: Three representative cases of non-reciprocal trans-port: (a) γ = 0 where NR = 17 . dB ; (b) γ = 0 . NR = 42 . dB ; and (c) γ = 0 .
18 where NR = 30 . dB . Themaximum non-reciprocity is observed in the domain around ω s and it is non-monotonic with respect to γ . C. Theoretical Analysis
The behavior of
N R ( γ ) seen in Fig. 4b can be under-stood within the framework of temporal coupled-modetheory [39]. Our calculation scheme breaks down the ef-fect of the magnetic field into two parts. First we considerthe effect to the resonant frequency of the individual res-onators separately (for γ = 0) in a magnetic substrate.For our applied field H it can be directly estimated fromFig. 2 to be ω ≈ . − . Next we add the effectof gain and loss γ in each of these resonances which arenow considered as a two level system and coupled via anon-magnetic substrate with a coupling constant Ω (i.e.evaluated with H = 0). This is estimated, to a good ap-proximation, from the eigenmode analysis of the isolatedset-up with H = 0 only in the domain between the tworesonators (see Fig. 4a), and is found to be Ω ≈ . − . The resulting symmetric ω (0) s and antisymmetric ω (0) a resonant modes of the isolated composite structureis then: ω (0) s/a = ω ∓ (cid:113) Ω − ( ργ ) (4)where ρ ≈ .
445 ns − is a scaling parameter that is ex-tracted from the analysis of the isolated set-up of Fig.4a. For this set up, the EP is γ PT = Ω /ρ ≈ . . The second part of our analysis considers the conse-quences of the magnetic field in the coupling between ω (0) s/a . Specifically, we consider that the resonances ( ω (0) s/a )are coupled via the magnetized substrate between thetwo microstrip cavities and indirectly via the presence ofthe bus wave-fields. In general, this additional couplingconstant λ is a function of the geometric properties ofthe two stripline resonators, the applied magnetic field, H , and the wavenumber k x of the bus field. Based onsymmetry considerations [33] we have that up to a linearapproximation, λ = λ + ı ( b k x + c H ) where λ , b , c are real parameters. When an incident electromagneticradiation with frequency ω in the vicinity of one of thesetwo resonances enters the bus waveguide, in either direc-tion, it will primarily excite the closer mode in frequencywithout being (to a good approximation) affected by thepresence of the other resonance. Below we consider thecase ω ≈ ω s where maximum non-reciprocity is observed.Therefore we will assume that the incident wave is cou-pled directly only with the symmetric mode.Under these assumptions, the temporal evolution ofthe symmetric ( a s ) and antisymmetric ( a a ) modal am-plitudes is described by the following equationsd a s dt = ıω (0) s a s − τ a s − λ ∗ a a + κ S in + κ S in d a a dt = ıω (0) a a a + λa s (5) S out − = S in − κ ∗ a s ; S out + = S in − κ ∗ a s where τ = τ − + τ + is the radiative coupling of the sym-metric mode to a left-going ( τ − ) or a right-going ( τ + )output wave, and { κ , κ } indicate the coupling con-stants between the symmetric mode and the incoming oroutgoing waves. We have that | κ | = τ + and | κ | = τ − .The modal amplitudes are normalized in such a way that | a s | ( | a a | ) correspond to the energy stored at the spe-cific mode, while (cid:12)(cid:12) S in (cid:12)(cid:12) and (cid:12)(cid:12) S in (cid:12)(cid:12) ( (cid:12)(cid:12) S out − (cid:12)(cid:12) and (cid:12)(cid:12) S out + (cid:12)(cid:12) )are the powers carried by incoming (outgoing) waves from(to) two different directions of the bus waveguide.The forward T F ≡ | S out + | | S in | and backward T B ≡ | S out − | | S in | transmittance for a left S in ∝ e ıωt and right S in ∝ e ıωt incident monochromatic field can be calculated from Eq.(5) by imposing the appropriate boundary conditions S in = 0 and S in = 0 respectively. We obtain that T F / B ( ω ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i (cid:18) ω − ω (0) s − | λ F / B | ( ω − ω (0) a ) (cid:19) ∓ ∆ (cid:15)i (cid:18) ω − ω (0) s − | λ F / B | (cid:16) ω − ω (0) a (cid:17) (cid:19) + (cid:0) τ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (6)where ∆ ε = τ + − τ − (cid:54) = 0 due to gyrotropy and λ F/B is the coupling between ω (0) s and ω (0) a for forward andbackward propagation.From Fig. 3 we observe that the maximum N R occursat the resonance frequency ω Bs of the BWD propagationwhich can be estimated from Eq. (6) to be ω B s = ω − (cid:112) Ω − ( ργ ) . The dependence of ω Bs on H allows usto reconfigure the position of maximum non-reciprocity.The modified coupling Ω ≡ (cid:113) Ω + | λ B | is a result ofthe external magnetic field which now also acts at thesubstrate between the two cavities and the presence ofthe incident wave in the bus waveguide. It allows usto estimate the gain and loss parameter γ ˜ PT = Ω /ρ for N R ( d B ) Gain/Loss parameter g R e ( w ) ( n s e c - ) (a) I m ( w ) ( n s e c - ) (b) g N R ( d B ) FIG. 4: (a) We show the dependence of simulated resonantmodes ( R e ( ω ) , • / I m ( ω ), (cid:4) ) on the gain/loss parameter γ for the set-up with H = 0 only in the domain between thetwo micro-cavities. A fitting using Eq. (4) (solid line) gives ω ≈ . ≈ . ρ ≈ .
445 (all measured innsec − ) corresponding to γ PT ≈ . T from the simulations ( • ) and fromthe theoretical expressions Eqs. (6) ( ◦ ). The green line isobtained using Eq. (7). The inset shows the analogous simu-lated NR for a 15% reduction in the bias field. The verticaldashed line indicates the position of the EP in this case. which we have an EP singularity for the isolated systemwith the uniform magnetic field (see Fig. 2).These theoretical results compare nicely with theCOMSOL simulations in Fig. 3 in the domain of ω ≈ ω Bs .A non-linear least square fit has been used in order to fitEq. (6) to the data for T B . The parameters that we haveobtained are ∆ (cid:15) ≈ − . τ ≈ . η ≈ . × − (all measured in nsec − ) and | λ B | ≈ .
111 nsec − . Allthese parameters, apart from | λ F | , have been kept fixedfor the forward transmission T F , see Eq. (6). The fittingvalue of T F indicated that | λ F | ≈ | λ B | nsec − . Finally,using Eqs. (6) together with Eq. (3) we have calculated N R versus γ . These theoretical results are shown in Fig.4b together with the simulations of COMSOL.In order to enhance our understanding of the originof the giant nonreciprocal effect we have further ap-proximated N R at ω = ω Bs . Guided by the numer-ics, which indicates that T F ( ω Bs ) ∼ O (1) in this fre-quency domain, we have assumed that log T F (cid:0) ω B s (cid:1) isnegligible when compared to log T B (cid:0) ω B s (cid:1) . Therefore N R ( γ ) ≈ | log T B ( ω B s ) | . This approximation leads usto the following expression up to leading order in η, ∆ (cid:15) and (cid:15) ≡ / (2 τ ) [40]:NR =
20 log
10 1+ εη (cid:16) √ β √ β (cid:17) ∆ ε η (cid:16) √ β √ β (cid:17) ; 0 < γ < γ PT
10 log
10 ( η + ε ) + βη ( η +2 ε )( η +∆ ε/ + βη ( η +∆ ε ) ; γ PT < γ < γ ˜ PT (7)where β ≡ Ω − ( ργ ) | λ B | .A further analysis of Eq. (7), indicates that when ∆ ε η < min (cid:110) − ΩΩ +Ω , − ε ε + η (cid:111) , then N R ( γ ) has a sin-gle maximum in the exact phase i.e. 0 ≤ γ ≤ γ ˜ PT ( H ) which occur at some critical value γ = γ NR .In case ∆ ε η < −
1, we have γ NR = γ PT while for − < ∆ ε η < min (cid:110) − ΩΩ +Ω , − ε ε + η (cid:111) we have γ NR = (cid:114) ( γ PT ) − | λ B /ρ | ( ∆ ε η / ( ∆ ε η )) − . Thus we conclude that theexistence and position of γ NR is strongly dictated by γ PT and | λ B | , i.e., this giant non-reciprocal behavior is a con-sequence of an interplay between the EP degeneracy andthe interaction of fields within the gyrotropic substrate. III. LUMPED CIRCUIT ANALYSIS
The EP-induced giant non-reciprocity can be furtheranalyzed utilizing an electronic circuit analog that main-tains the essence of the original physics while also allow-ing a significantly simplified path toward both analyticand numeric analysis. The circuit, shown in Fig. 5(a),reduces the parallel microstrip resonators to a pair of
RLC resonators capacitively coupled to points separatedby a distance d along an ideal TEM transmission line.The inter-resonator coupling through the gyrotropicallyactive substrate is incorporated as a mutual inductance M in parallel with an ideal gyration G such that theinductor currents are related to the voltages by (cid:18) I I (cid:19) = 1 iω (cid:20) L MM L (cid:21) − (cid:18) V V (cid:19) + (cid:20) G − G (cid:21) (cid:18) V V (cid:19) (8)The gain and loss, along with the small inherent loss η defined earlier, are implemented by negative and positiveparallel resistances of slightly different magnitude.In the frequency domain, Kirchoff’s Laws for this cir-cuit are easily expressed, though transcendental due tothe trigonometric wave components in the center trans-mission line section. All seven element of the circuit ( L , C , R , R , M , G , C c , and d ) represent essential featuresof the original structure that can contribute to the en-hancement of the transmission nonreciprocity. Note that G plays a similar role as the static magnetic field H inthe gyrotropic substrate of the microstrip device and isthe key circuit element responsible for nonreciprocity.The main graphs of Fig. 5(b)-(d) illustrate numeri-cal results exploring the NR with gain/loss and gyrationstrengths, γ = ( R − + R − ) (cid:112) L/C respectively, to a detail that is computationally expensive in the COM-SOL simulation, and somewhat abstract in the theoret-ical analysis. The NR density plot shown in Fig. 5(b)is separated into two regions by the black solid line rep-resenting the position of the isolated exceptional point,with the exact ˜ PT phase above and the broken phasebelow. The singular NR is seen as the bright swath within the unbroken region just above [14]. Figure 5(c)show cuts of the NR at several fixed values of the gy-ration strength g (below) along with the correspondingisolated dimer eigenfrequencies (above). Note again thatthe maximum NR occurs below the isolated exceptionalpoints. The similarities with Fig. 4 associated with thephotonic structure is striking, thus indicating the sharedNR mechanism. Specifically for γ = 0 we again observe amoderate non-reciprocal behavior which is dramaticallyenhanced at γ -values close to γ ˜ PT . This can be betterappreciated by analyzing the parametric evolution of theeigenfrequencies of the isolated circuit. The isolated sys-tem in this electronic analog includes all of the effects ofthe resonator coupling, such as the gyration, fulfilling theinequality expressed in Eq. 7. IV. DISCUSSION
We have shown that the flexibility introduced by the˜ PT properties of the photonic resonator dimer dramat-ically enhance the strength – and hence the bandwidth– of the singular nonreciprocity. We have observed thisover certain ranges of the system parameters. At thesame time we have demonstrated that these results ap-ply equally well in the case of lumped circuitry.This universal nature of the giant non-reciprocal re-sponse near the EP calls for an intuitive explanation.First we have to realize that the structure constitutes aneffective ring since the two cavities are directly coupledto one-another while at the same time they are coupledindirectly via the bus waveguide. At the EP the two su-permodes of the effective ring structure are degeneratehaving a definite chirality [12]. The presence of the mag-netic field breaks the spectral degeneracy, while weaklypreserving the (common) chiral nature of the modes. Asa result the two modes are coupled differently with aleft and a right incident wave. Assuming, for examplein the electronic set up of Fig. 5(a), that the chiralityof the modes is clockwise (CW) we conclude that due tophase matching such a mode will be coupled only to a leftincident wave but not to a right incident one. Accord-ingly, the left incident wave will excite the CW super-mode while at the same time can exploit a direct opticalpath associated with a transmission via a direct processbetween the incident and transmitted channels. Theseoptical paths can interfere destructively at the outputchannel (depending on the propagation phase associatedwith the length of the bus waveguide and the gyrotropy)leading to a Fano effect and consequently to a (near) zerotransmittance. An important condition here is that the -0.500.5 0.2 0.4 0.60204060 g=.1g=.25g=.4g=.55 g R e ( ω / ω ) - N R ( d B ) I m ( ω )[ / √ ( L C )] g γ (a)(b) (d)(c) FIG. 5: Exploration of the gain/loss γ = ( R − + R − ) (cid:112) L/C and gyration strength g = G (cid:112) L/C parameter space of thelumped circuit model shown in (a). In (b) we plot the nonreciprocity NR as intensity (high values of NR correspond to brightareas while low values of NR to dark areas) in the map. Due to the limitation of the resolution, the narrow peaks representinghigh NR ( >
30 dB) in Fig. 5 (c) are not resolved by the color bar. (c) We show some indicative ”cuts” from the densitymap at several gyration strengths (shown in (b)) for Z (cid:112) C/L = 0 . kd ≈ π at the LC resonant frequency, C c /C = 0 . M/L = 0 .
03, and η = ( R − − R − ) (cid:112) L/C = 0 .
03 for the intrinsic loss. (d) Shows the corresponding real and imaginary partsof the balanced, isolated ( η = C c = 0) dimer mode frequencies illustrating relation of the exceptional points to the singularitiesof the giant non-reciprocity. The solid line through the NR density plot shows the position of the isolated system exceptionalpoint, slightly beyond the singularity. internal losses of the cavities are small so that the two in-terfering waves have the same amplitudes. On the otherhand, a right incident wave, because of phase mismatch,does not couple to the CW chiral supermode of the ef-fective ring. As a result it does not experience the inter-nal losses inside the cavity and consequently the (direct)transmission is high.The electronic circuit that we proposed can be real-ized experimentally using existing MOSFET technolo-gies. Such reconfigurable circuitry (due to on-the-fly ma-nipulation of gain and loss) would be useful in the real-ization of RF circulators and isolators. Moreover in theoptical domain where the magneto-optical effects are veryweak, the wave propagation can be masked by unwantedlosses associated with the materials used as a means torealize non-reciprocal propagation. Our scheme – withthe manipulated gain and loss – would resolve some ofthe above mentioned issues and help restore a strong non-reciprocal signal. V. CONCLUSIONS
We have theoretically defined the conditions for whicha non-Hermitian structure with antilinear symmetry can lead to giant non-reciprocal transport: the system hasto operate in the stable domain and in the vicinity ofan EP singularity which amplify the effects of gyrotropy.The non-reciprocal frequency domain is reconfigurable,albeit is narrow-band. We have demonstrated the valid-ity of the theoretical predictions in the microwave domainwhere we have found non-reciprocal transmission whichis higher than 90dB. We have further confirm the gener-ality of our results utilizing a user-friendly framework oflump circuits. It will be interesting to extend this studyand investigate giant non-reciprocal transport in acousticor matter-wave systems where amplification and atten-uation mechanisms can be easily controlled and used torealize EP degeneracies [41, 42] while an effective mag-netic field can be introduced via time-varying potentials[43, 44].
Acknowledgements -
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