Abstract
Timeasymmetric stateevolution properties while encircling an exceptional point are presently of great interest in search of new principles for controlling atomic and optical systems. Here, we show that encirclinganexceptionalpoint interactions that are essentially reciprocal in the linear interaction regime make a plausible nonlinear integrated optical device architecture highly nonreciprocal over an extremely broad spectrum. In the proposed strategy, we describe an experimentally realizable coupledwaveguide structure that supports an encirclinganexceptionalpoint parametric evolution under the influence of a gain saturation nonlinearity. Using an intuitive timedependent Hamiltonian and rigorous numerical computations, we demonstrate strictly nonreciprocal optical transmission with a forwardtobackward transmission ratio exceeding 10 dB and high forward transmission efficiency (∼100%) persisting over an extremely broad bandwidth approaching 100 THz. This predicted performance strongly encourages experimental realization of the proposed concept to establish a practical onchip optical nonreciprocal element for ultrashort laser pulses and broadband highdensity optical signal processing.
Introduction
Nonreciprocal light propagation is the key performance attribute of optical isolators and circulators, elements essential for optical signal processing, telecommunications and the protection of highpower laser systems. Whereas the Faraday effect in magnetooptic crystals enables broadband and highefficiency nonreciprocal elements for freespace systems, the realization of integrated onchip nonreciprocal elements remains elusive. Different approaches have been proposed to realize integrationcompatible, onchip nonreciprocal devices. To this end, indirect photonic transitions mediated by dynamic index modulation^{1,2} and nonlinear resonance shifts in asymmetric highQ microcavities^{3,4,5} have been studied in depth. These effects produce remarkable nonreciprocal transmission within a submm or even μmscale device footprint; however, other performance attributes are problematic such as a narrow bandwidth, low forwardtransmission efficiency and a high signal power threshold for operation. These problems must be eliminated or at least significantly alleviated to process broadband signals (including modulation) to respect tight energyconservation constraints imposed by applications and for stable operation in ambient conditions involving moderate temperature drifts.
Optical nonreciprocity in nonlinear paritytime (PT) symmetric systems^{6,7,8} are presently attracting considerable attention. In PTsymmetric coupled microcavities, optical modes undergo a spontaneous symmetrybreaking transition at an exceptional point (EP). Entering further into the brokensymmetry phase leads to enhanced cavityexcitation asymmetry between two opposite coupling directions. Including auxiliary nonlinear effects, such as the optical Kerr effect and gain saturation^{6,7}, strong nonreciprocal transmission is obtained as opposed to strictly reciprocal PTsymmetric effects in the linear and stationary systems^{9,10,11,12,13,14,15,16}. In spite of high nonreciprocal transmission ratios and low signal power thresholds for operation, the functionality of such approaches is available only over an ultranarrow bandwidth on the order of MHz and involves unpredictable laser oscillations causing critical instability. These problems are difficult to avoid in highQ resonatorbased approaches.
In this paper, we propose an integrationcompatible, nonresonant broadband nonreciprocal device concept inspired by nonHermitian quantummechanical interactions near an EP. We employ a timevarying nonHermitian Hamiltonian along a parametric path that encircles an EP where the canonical quantum adiabatic theorem breaks down exclusively for one preferred temporal direction. A spatial analogy of this effect is manifested in a nonlinear coupledwaveguide structure with amplifying and attenuating waveguides. Importantly, the obtained asymmetric optical propagation is totally unrelated to interferometric power beating or any resonant optical excitations that result in a strong wavelength dependence. Thus, insensitivity to the excitation wavelength is obtained as opposed to previously proposed schemes. Including normal gain saturation and the consequent power regulation effect yields robust nonreciprocal transmission over an extremely broad spectral band that is only limited by the gain bandwidth of the gain material selected.
Results
Design and basic performance
We consider a waveguide architecture shown in Fig. 1a. The system comprises a unidirectional mode converter section implemented as nonlinear coupled waveguides connected to input and output Ybranches and singlemode input and output waveguides. The unidirectional converter is a key functional region where oneway adiabatic modal transformation occurs. For forward (lefttoright) propagation, an incident mode from the input singlemode waveguide and Ybranch preserves its transversal symmetry during propagation over the converter region, and thus freely transmits to the output waveguide. For backward (righttoleft) propagation, an incident symmetric mode is converted into an antisymmetric output mode that is eventually rejected at the Ybranch because of modal incompatibility with the singlemode waveguide.
To produce the envisioned nonreciprocal transmission, we apply complex effective index profiles in the coupledwaveguide section, as indicated in Fig. 1b. The effective indices n_{1} and n_{2} of the fundamental guided modes in the upper and lower arms, respectively, are defined as n_{p}=β_{p}/k_{0}=n_{c}+Δn_{p}(z), where p takes on 1 for the upper attenuating waveguide or 2 for the lower amplifying waveguide, β_{p} is the propagation constant of the fundamental guided mode in waveguide p, k_{0} is the vacuum wavenumber and n_{c} is the average real effective index. Here, Δn_{p}(z) is the desired complex modulation profile given by
where L is length of the converter region and S(I_{2})=(1+I_{2}/I_{S})^{−1} is a gain saturation factor with saturation intensity constant I_{S} and local intensity I_{2} in waveguide 2. We note that the specific Δn_{p}(z) profiles described by equations (1) and (2) are a representative case among a wide variety of other possible profiles, as we will explain in the next section.
Figure 1c–e summarizes twodimensional finite element method calculations assuming singlemode slab waveguides with Δn_{0}=4.26 × 10^{−4}, L=5 mm, a waveguide core width of 1 μm, a core separation distance of 2 μm and an operating wavelength of 1.13 μm (other parameters and conditions are given in the figure caption). For an isolated waveguide of these parameters, the maximum linear modal gain/attenuation constants are ±411.4 dB cm^{−1} and the total amplification/attenuation over the 5mmlong unidirectional converter region are ±102.8 dB. These levels of optical gain and loss are readily obtainable with a variety of optical gain materials such as conventional direct bandgap semiconductors and dyedoped polymers. We observe in Fig. 1c,d that forward propagation yields significantly amplified transmission at the righthand singlemode waveguide output, whereas most of the output mode energy diverges into the cladding for backward propagation, thereby transmitting a negligibly low optical power. The output intensity profiles in Fig. 1e for the forward and backward propagation cases clearly demonstrate highquality nonreciprocal transmission ratio (NTR) of 28.7 dB and amplified forward transmission with a gain of +4.95 dB. This nonreciprocal effect in a stationary and nonresonant system is caused by a nonlinear nonHermitian wave interaction near a PTsymmetric EP as we will explain in the next section.
Principle and fundamental properties
We describe the dynamics of optical modes over the converter region using the coupledmode formalism:
where κ is the coupling constant and A_{p} denotes the amplitude of the fundamental mode in waveguide p. We write the frequencydomain total electric field of the coupled section as E(x,y,z)=[A_{1}(z)E_{1}(x,y)+A_{2}(z)E_{2}(x,y)]·exp(in_{c}k_{0}z) with E_{p}(x,y) indicating the normalized wavefunction of the fundamental mode in waveguide p. In the lowintensity limit where the gain saturation factor S(I_{2})≈1, equation (3) can be expressed as a linear Schrödingertype equation dψ_{fw}(t)>/dt=i H(t)ψ_{fw}(t)> with an effective Hamiltonian
where we define the forward dynamic state vector ψ_{fw}(t)>≡[A_{1}(t) A_{2}(t)]^{T}, the fictitious time variable t≡κk_{0}z and the reduced energy parameter ξ(t)≡Δn_{1}(t/κk_{0})/κ. The parametric spectra of eigenvalues λ_{±}=±(1+ξ^{2})^{1/2} of H on the complex ξ plane have characteristic singularities at ξ=±i corresponding to a pair of PTsymmetric EPs. A comprehensive description and the general features of nonHermitian singularities of this kind are found in ref. 17. On the complex ξ plane, the effective index modulation profiles given by equations (1) and (2) imply circular trajectories around the EP for Δn_{0}>κ/2, as shown in Fig. 2a. For the simulated case in Fig. 1c–e, ξ(t) encircles the EP at ξ=+i counterclockwise. The backward propagation is described by a timereversal transformation t→T–t and ξ(T–t) encircles the EP clockwise, where T=κk_{0}L is total (fictitious) time duration for a single parametric revolution. Time evolution of the backward dynamic state ψ_{bw}(t)> is thus governed by the timereversed Schrödingertype equation dψ_{bw}(t)>/dt=–i H(T–t)ψ_{bw}(t)>.
For our timevarying Hamiltonian H(t), the instantaneous eigensystem, determined by a local eigenvalue equation H(t)φ_{μ}(t)>=λ_{μ}(t)φ_{μ}(t)>, reveals a typical complex square root distribution as plotted in Fig. 2b. Rigorously defining the instantaneous eigenvalue–eigenvector pair {λ_{μ}(t), φ_{μ}(t)>} such that they are continuous functions of t for 0≤t≤T, we use a branch cut at Re(ξ)=0 for –1≤Im(ξ)≤1. We show λ_{μ} surfaces on the complex ξ plane and two trajectories for λ_{μ}(t) in Fig. 2b. The eigenvalue surface is divided into two sheets by the branchcut demarcation. A sheet representing λ_{G} (λ_{G} sheet) has a negative imaginary part as indicated by the white–blue–cyan skin and the other sheet for λ_{L} (λ_{L} sheet) has a positive imaginary part as indicated by the white–red–yellow skin. For ξ(t) corresponding to equations (1) and (2), λ_{G}(t) and λ_{L}(t) appear as spiral curves on the λ_{G} and λ_{L} sheets, respectively.
The nature of the corresponding instantaneous eigenvectors φ_{G}(t)> and φ_{L}(t)> are understood to represent amplifying (gain) and attenuating (loss) modes, respectively. We note at the start and end points (t=0 and T) that ξ=0 and the instantaneous eigenvectors φ_{G}> and φ_{L}> are either an even mode, even>=2^{−1/2} [1 1]^{T}, with the eigenvalue λ=+1 or an odd mode, odd>=2^{−1/2} [1–1]^{T}, with λ=–1. In particular, φ_{G}(0)>=φ_{L}(0)>=even> whereas φ_{L}(T)>=φ_{G}(T)>=odd>. Therefore, φ_{G}(t:0→T)> and φ_{L}(t:0→T)> continuously evolve from even> into odd>. These mode swapping evolution passages of the instantaneous eigenvectors around the PTsymmetric EP is often referred to as an adiabatic state flip. The adiabatic state flip and associated geometricphase accumulation were experimentally confirmed in coupled microwave cavities^{18,19} and exciton–polaritonic quantum billiard experiments^{20}. These adiabatic interaction properties have been confirmed by tracking the instantaneous eigenstates in the quasistationary limit. The complete dynamics of state evolution in nonHermitian systems with significant imaginary eigenvalue splitting in general involve highly nonadiabatic behaviour associated with an antiadiabatic state jump occurring under appropriate initialstate and parametric conditions^{21,22,23,24}. The antiadiabatic state jump is a key interaction leading to the timeasymmetric stateevolution passages in our proposed system.
The essence of the antiadiabatic state jump and associated timeasymmetric effects is revealed in a case where the initial state is given by either one of the instantaneous eigenvectors, that is, ψ(0)>=φ_{μ}(0)>. In the limit <φ_{μ}ψ> >> <φ_{ν}ψ> that is likely for a slowly varying system satisfying the wellknown quantum adiabatic condition^{22} <H/t> << λ_{μ}–λ_{ν}^{2}, the probability amplitude for nonadiabatic transition from φ_{μ}> to φ_{ν}> is approximated by:
where g_{νμ}(t)=<φ_{ν}*dH/dtφ_{μ}>/(λ_{ν}–λ_{μ}) is a nonadiabatic coupling constant and is average eigenvalue over the time domain [τ, t]. Note here that the inner product <··> of two state vectors is the cproduct following the biorthogonal treatment for nonHermitian systems^{25,26}. See Supplementary Note 1 for the detailed derivation. For Hermitian Hamiltonians that essentially involve purely real eigenvalues λ_{μ} and λ_{ν}, C_{νμ}(t) remains negligible for slowly varying H(t) under the quantum adiabatic condition. Consequently, the time evolution of the state ψ(t)> in a Hermitian system simply follows the instantaneous eigenvector passage φ_{μ}(t)>.
However, in nonHermitian cases where the eigenvalues include significant imaginary parts, the standard quantum adiabatic theorem fails to properly describe the state evolution. A radical breakdown of the standard quantum adiabatic theorem for the forward propagation state ψ_{fw}(t)> is seen in Fig. 3a as an antiadiabatic jump of the expectation value <H(t)>_{fw}≡<ψ_{fw}H(t)ψ_{fw}>/<ψ_{fw}ψ_{fw}> from the λ_{L} sheet to the λ_{G} sheet, in stark contrast to the highly adiabatic expectation value passage <H(t)>_{bw}≡<ψ_{bw}H(t)ψ_{bw}>/<ψ_{bw}ψ_{bw}> for the backward state in Fig. 3b. Therein, the forward expectation value <H(t)>_{fw} passage under the initial condition <H(0)>_{fw}=+1 follows the λ_{L} sheet in the beginning, undergoes the antiadiabatic jump towards the λ_{G} sheet and eventually ends up at <H(T)>_{fw}=+1. This forward expectation value passage indicates the sequential mode transition of even>→φ_{L}(t)>→φ_{G}(T–t)>→even> involving the antiadiabatic jump from φ_{L}> to φ_{G}>. In the numerical calculation of Fig. 1c, manifestation of the antiadiabatic jump appears as a transition of the modal intensity profile from an initially attenuating pattern to an amplifying pattern in the region indicated by a white dotted box. Considering general aspects of the antiadiabatic jump, equation (5) implies that any evolution passage on the λ_{L} sheet undergoes the antiadiabatic jump towards the λ_{G} sheet whenever the dwell time on the λ_{L} sheet exceeds a critical time interval T_{c}≡Im(Λ_{L}–Λ_{G})^{−1}. A conceptually equivalent time parameter to T_{c} was noted by Milburn et al.^{27} as a time of stability loss delay for smallradius encirclinganEP parametric paths.
In the backward propagation case, the expectation value <H(t)>_{bw} passage indicates a modeswapping adiabatic evolution that follows a simple process of even>→φ_{G}(t)>→odd> as shown in Fig. 3b. In particular, ψ_{bw}(t)> follows the instantaneous eigenvector φ_{G}(t)> for H(t) quickly varying in time even beyond the standard quantum adiabatic condition. This type of superadiabatic evolution passage is expected for any dynamic state under the condition ψ>≈φ_{G}> for which the expectation value <H> appears on the λ_{G} sheet. In Supplementary Note 1, we explain in greater detail the fundamental reasons for the antiadiabatic jump and the superadiabatic evolution passages, depending on the direction of timeevolution and the initial conditions.
Looking at the timedomain profiles of the channel amplitude ratio A_{2}/A_{1}, the dynamic properties of the state evolution associated with the forward antiadiabatic and backward superadiabatic passages are more clearly observable. We show the real and imaginary parts of A_{2}/A_{1} for ψ_{fw}> and ψ_{bw}> as functions of t in Fig. 3c. The forward passage ψ_{fw}(t)> undergoes the following sequential behaviour from t=0 to t=T=25: it is launched with A_{2}/A_{1}=1 corresponding to state even>; slowly deviates with a spiral oscillation from φ_{L}> to t≈4; suddenly diverges from φ_{L}> at t≈T_{c}≈6, indicating the antiadiabatic jump; converges to φ_{G}> at t≈10; and finally ends up in state even> through the exact adiabatic φ_{G}> passage. In contrast, the backward passage of ψ_{bw}(T–t)> from the initial state even> simply follows the adiabatic passage of φ_{G}> to eventually end up at state odd>. Interestingly, a spiral oscillation of ψ_{bw}(T–t)> because of nonadiabatic coupling over the domain of 20≤T–t≤25 vanishes within T_{c}≈6, indicating the superadiabatic evolution property.
Antiadiabatic properties and associated timeasymmetric effects have been studied previously. The inevitability of timeasymmetric antiadiabatic jumps in nonHermitian systems has been theoretically argued in the case of molecular vibrational state transfer because of chirped pulses^{22,23} and for dualmode optical waveguides^{28}. Very recently, this seemingly counterintuitive effect was experimentally verified independently in a microwave channel waveguide structure^{29} and in an optomechanical system^{30}. In previous studies, the circular geometry for parametric trajectories was not the unique case and arbitrary geometrical paths enclosing an EP could be used for inducing the required timeasymmetric antiadiabatic jump.
The time asymmetry in the stateevolution passages provides a key principle for our proposed nonreciprocal device concept. This property by itself is not yet sufficient to accomplish strictly nonreciprocal photonic transmission as the fundamental Lorentz reciprocity theorem for linear electromagnetism in stationary optical systems requires^{15,16}. Hence, we introduce a gain saturation nonlinearity to break the strict reciprocity in transmission. In Fig. 4a,b, we show evenmode power ratio P_{sym}/P_{tot} profiles for the forward and backward passages in the linear (I_{S}=∞) and nonlinear (I_{S}=I_{0}) cases, respectively. Therein, we find negligible differences in the timeasymmetric stateevolution properties between the linear and nonlinear cases. The key properties such as the paritypreserving forward evolution (unity order P_{sym}/P_{tot} value at t=25) with an antiadiabatic statejump signature at t≈T_{c}≈6, and the parityexchanging, adiabatic backward evolution (low P_{sym}/P_{tot} value at t=25) persist for the highly nonlinear case in the almost identical manners to those for the linear case. In a quantitative comparison, the differences in the final P_{sym}/P_{tot} values between the linear and nonlinear cases are below 0.02%. The role of the gain saturation nonlinearity is to utilize the timeasymmetric evenmode power ratio as the nonreciprocal transmission ratio. In Fig. 4c, we show the forward and backward total power P_{tot}(t) profiles in the linear (I_{0}<<I_{S}=∞) and nonlinear (I_{0}=I_{S}) cases. In the final state at t=25, the forward and backward P_{tot} values in the nonlinear case are identical to each other at P_{tot}=4.8P_{0} (log_{10}(P_{tot}/P_{0})=0.681). This is because the gain saturation nonlinearity equalizes the total power once the modal intensity level in the amplifying arm becomes significant with respect to the saturation level of I_{S}. Assuming that only the even mode contributes to the final transmission toward the output singlemode waveguide, the nonreciprocal transmission ratio is identical to the ratio of the forward evenmode power ratio to the backward evenmode power ratio.
In the linear case, however, the final forward P_{tot} value is substantially lower than the final backward P_{tot} value as shown in Fig. 4c, implying that the timeasymmetric evenmode power ratio does not yield a significant nonreciprocity. In particular, P_{tot} for the backward passage ψ_{bw}> grows monotonically as the state simply follows the amplifyingmode φ_{G}(T–t)> passage. In contrast, P_{tot} for the forward passage ψ_{fw}> is strongly attenuated for its transient dwell time in the attenuatingstate φ_{L}(t)> passage before the antiadiabatic jump occurs at t≈T_{c}≈6. This difference in the poweramplification history results in the observed difference in P_{tot} depending on the propagation direction. Essentially, the subtle processes of nonadiabatic coupling, amplification and transient attenuation in the linear regime render the partial evenmode powers P_{sym} for ψ_{bw}> and ψ_{fw}> at t=T exactly identical to each other. Therefore, although the timeasymmetric stateevolution passages induce large difference in the evenmode power ratio P_{sym}/P_{tot} for the two opposite encirclinganEP directions, the reciprocity in the transmission is unbroken in the linear case.
As discussed earlier for the strong gain saturation regime, in which the total power equalization effect is fully realized, the NTR is given by the ratio of the forward evenmode power ratio to the backward evenmode power ratio. Following this argument, under the condition of <evenψ_{fw}(T)>^{2}≈<ψ_{fw}(T)ψ_{fw}(T)>^{2}≈<ψ_{bw}(T)ψ_{bw}(T)>^{2}, the NTR takes on a simple form
as approximated from equation (5) for T>>1. Here, the coefficient α is determined by the geometry of ξ(t) on the complex ξ plane. In equation (5) for t=T, a partial integration domain capturing a significant contribution of the instantaneous nonadiabatic coupling amplitude g_{LG}(τ) to C_{LG}(T) corresponds to T–T_{c}<τ<T. Taking this effective domain for the integration in equation (5), an approximate expression is found such that C_{LG}(T)≈f(T)·T_{c}/T, where f=(ξ/ζ)·<φ_{L}H/ξφ_{G}>·(λ_{L}–λ_{G})^{−1} with ζ=z/L. Importantly, equation (6) implies that the NTR is determined by the purity of the backward dynamic state ψ_{bw}> with respect to the amplifying eigenvector passage φ_{G}>. The NTR is not affected by the phase difference between the eigenmodes or by interferenceinduced power beating effects that are highly sensitive to the operating wavelength and device length in general. In Fig. 5, we show the dependence of R_{NTR} on T for several values of I_{0}/I_{S}. The T^{2} dependence of the NTR at large T is confirmed quantitatively. As implied in equation (6), the R_{NTR}(T) profiles have no periodic feature that would normally be associated with interferenceinduced power beating on the scale of the conventional beat length ΔT=1, that is, ΔL=(κk_{0})^{−1}. This property is a unique feature of our proposed concept enabling broadband optical nonreciprocity.
We evaluate the spectral characteristics of the NTR and forward transmission efficiency (FTE) as major performance parameters. In the spectral analysis, we recall the model device used in Fig. 1c–e and calculate the NTR and FTE spectra for L=1, 5 and 10 mm, whereas other structural parameters and optical constants remain identical to those indicated in the caption of Fig. 1. We use the classical Runge–Kutta (RK4) method for solving the nonlinear coupledmode model of equations (1, 2, 3). The results are given in Fig. 6. Quantitative agreement between the spectral curves obtained from the nonlinear coupledmode model and the symbols obtained from the fully vectorial finiteelement method confirms the validity of our theoretical approach. Major performance parameters estimated from the data in Fig. 6 are summarized in Table 1. When compared with the 1to100 GHz Δν_{10dB} of resonatorbased optical isolators^{3,4,31} and the 1 THz Δν_{10dB} of dynamic refractive index modulation approaches^{1,2}, the bandwidths over which our NTR remains high (> 10 dB) and our FTE near unity are remarkably high, exceeding 100 THz.
The essential underlying mechanisms for the strong nonreciprocal property are the EPinduced asymmetry in the evenmode power ratio and the total power equalization effect because of the gain saturation nonlinearity. These two effects have different spectral properties, leading to characteristic spectral features such as a bellshaped profile in the NTR and a thresholdlike behaviour in the FTE near the wavelength of 1.18 μm, as observed in Fig. 6. First, according to the argument leading to equation (6), the EPinduced asymmetry in the evenmode power ratio is a quadratic function of the total evolutiontime parameter T=κk_{0}L that is monotonically increasing with wavelength because κ exponentially grows with wavelength as determined by a field overlap between the guided modes in waveguides 1 and 2. In contrast, the gain saturationinduced total power equalization effect becomes weaker in the longer wavelength domain as the parametric ξ(t)=Δn_{1}/κ trajectory in the shorter wavelength range enters deeper into the highly nonHermitian domain above the EP where large imaginary eigenvalue splitting results in a rapid power amplification of the gain mode φ_{G}(t)> passage—see the ξ(t) trajectory with respect to the EP on top of Fig. 6. Therefore, as determined by the tradeoff between the increasing EPinduced asymmetry in the evenmode power ratio and decreasing gain saturationinduced total power equalization effect with wavelength, the spectral region of maximum NTR appears in the intermediate wavelength region where the ξ(t) trajectory closely encircles the EP, as numerically confirmed in Fig. 6a. In the short wavelength limit, near the wavelength of 0.6 μm where the ξ(t) trajectory is far away from the EP during the whole evolution passage, the EPinduced time asymmetry is not significant because of small T, yielding a small NTR. On the other hand, in the long wavelength limit, near the wavelength of 1.6 μm, the gain saturationinduced total power equalization effect is weak and the optical transmission becomes reciprocal regardless of the degree of time asymmetry in the evenmode power ratio.
In addition, the imaginary eigenvalue splitting has a thresholdlike behaviour near the EP, and the circular ξ(t) trajectory does not encircle the EP any more at wavelengths longer than 1.18 μm. This behaviour is responsible for the thresholdlike character in the FTE spectra in Fig. 6b. Interestingly, the fairly high NTR persists in the longer wavelength region where the circular ξ(t) trajectory excludes the EP. In this region, forward propagation involves two antiadiabatic jumps preventing symmetry exchange, whereas backward propagation undergoes a single antiadiabatic jump resulting in symmetry exchange from the even to odd mode. Explaining the spectral characteristics of the FTE, a main consideration is to derive a relation between the maximum output intensity and key factors such as gain saturation intensity, gain/loss coefficients and the parametric geometry of the ξ(t) trajectory. Although this problem is beyond the scope of this paper, it would provide important information for systematically optimizing the device for applications.
Discussion
Considering the experimental feasibility of the proposed nonreciprocal device concept, an important issue is to determine efficient approaches to synthesize the required complex effective index profiles to dynamically encircle an EP. Although various methods such as positiondependent doping or the deposition of gain and loss agents on appropriatelycoupled channel waveguides can be considered, we briefly introduce a lithographic approach that does not involve yetunestablished fabrication issues.
The approach is illustrated in Fig. 7. The unidirectional mode converter region consists of two coupledchannel waveguides with an adjacent side patch. In this design, the real effective index profile is created by a zdependent waveguidecore width w_{2}(z), whereas the imaginary effective index profiles are created via the zdependent gapwidth d_{s}(z) between the waveguide core and side patch. In particular, the side patch induces leakage radiation from the guided mode toward the adjacent side patch at a desired rate for a given w_{2}(z) value, thereby controlling the imaginary effective index in an independent manner. In Supplementary Fig. 1 and Supplementary Note 2, we present an example design based on dyedoped polymer waveguides. Therein, the optimized design produces the desired encirclinganEP parametric modal evolution with the essential attributes. Advantageously, the example design has geometrical parameters that are highly favourable for standard nanolithographic fabrication processes and yields a theoretical performance consistent with the twodimensional design assumed for Fig. 6.
Additional discussions of the operating bandwidth restrictions because of the gain bandwidth of the selected emitter species and a Kramers–Kronig relation between the real and imaginary indices are also provided in Supplementary Note 2. Importantly, the proposed design maintains the required complex effective index profiles even if the optical constants of the constituent materials drift, thereby ensuring a more stable performance than other potential approaches based on positiondependent doping of gain and loss agents.
In summary, we proposed a principle for onchip broadband optical nonreciprocity enabled by nonlinear EP dynamics. Employing judiciously interrelated optical gain and absorption distributions, the proposed device architecture and associated operating principles produce highquality onchip optical nonreciprocity over a spectral bandwidth exceeding 100 THz. The anticipated performance is notably distinctive from previous approaches.
Thus, it is of great interest to experimentally realize the proposed device idea utilizing available materials and nanophotonic structures. In optimizing practical designs, various geometries producing parametric paths that encircle an EP should be taken into account to maximize the operating bandwidth, NTR and FTE. Limitations in the experimental performance should be carefully investigated in consideration of power thresholds and bandwidth restrictions resulting from the chosen optical gain mechanism as well as fabrication imperfections. In addition, we note that strong optical confinement and field enhancement in nanoplasmonic and highindex semiconductor platforms may yield much smaller device footprints and a lower power threshold. In a broader perspective, we hope that our results stimulate extensive research on various nonHermitian optical effects and associated device applications.
Data availability
The data that support the findings of this study are available from the corresponding authors on request.
Additional information
How to cite this article: Choi, Y. et al. Extremely broadband, onchip optical nonreciprocity enabled by mimicking nonlinear antiadiabatic quantum jumps near exceptional points. Nat. Commun. 8, 14154 doi: 10.1038/ncomms14154 (2017).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
This research was supported in part by the Basic Science Research Program (NRF2015R1A2A2A01007553) and by the Global Frontier Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT & Future Planning (NRF2014M3A6B3063708).
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Y.C., C.H. and S.H.S. conceived the original concept and initiated the work. Y.C. and J.W.Y. developed the theory and model. C.H. and Y.C. performed numerical analyses. J.W.Y. and Y.C. organized the results. J.W.Y., P.B., Y.C. and S.H.S. wrote the manuscript. All authors discussed the results. Y.C. and C.H. contributed equally to this work.
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Choi, Y., Hahn, C., Yoon, J. et al. Extremely broadband, onchip optical nonreciprocity enabled by mimicking nonlinear antiadiabatic quantum jumps near exceptional points. Nat Commun 8, 14154 (2017). https://doi.org/10.1038/ncomms14154
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