Exceptional Points in Plasmonic Waveguides Do Not Require Gain or Loss
Sung Yoon Min ,1 Ju Young Kim,1 Sunkyu Yu ,2 Sergey G. Menabde ,1,* and Min Seok Jang 1,†
1School of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
2Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea
(Received 6 May 2020; revised 11 August 2020; accepted 5 October 2020; published 17 November 2020)
Exceptional points (EPs) in photonics are associated with non-Hermitian Hamiltonians with energygain or loss, a notable example is EPs in parity-time symmetric Hamiltonians. We show that, counterintu-itively, actual energy gain or loss is not required to generate this type of non-Hermitian degeneracy, sincethe eigenvalue of the optical Hamiltonian is the mode’s propagation constant and not the energy. This isdemonstrated by a simple three-layer insulator-metal-insulator (I -M -I ) plasmonic waveguide, the eigen-modes of which are known to experience degeneracy at a certain point in the parametric space suggestedto be used for rainbow trapping. We identify this point to be, in fact, an EP with the coalescence of theeigenmodes, despite the system having neither parity nor time symmetry. Furthermore, we demonstratethe manifestation of a third-order EP, which is generated by merging two separate EPs in the parametricspace of the I -M -I waveguide. The presented results reveal unconventional properties of the Hamiltonianof a simple plasmonic waveguide and provide an insight into the nature of EPs in non-Hermitian plas-monic systems in general, suggesting the possibility of accessing even higher-order EP regimes in simplephotonic structures without the need for optical gain or loss.
DOI: 10.1103/PhysRevApplied.14.054041
I. INTRODUCTION
Non-Hermitian Hamiltonians with complex eigenvaluesare widely used to describe physical systems with energyexchange, opposite to Hermitian Hamiltonians that signifyideal systems with energy conservation and real eigenval-ues. However, non-Hermitian Hamiltonians possess muchricher physics that was only recently unmasked. Thegroundbreaking study of Bender and Boettcher showedthat even systems described by non-Hermitian Hamiltoni-ans could exhibit real eigenvalues if the parity-time (PT)symmetry was satisfied [1]. These interesting notions ofPT symmetry have been recently embraced by the field ofphotonics [2], where several optically coupled subsystemscan exhibit gain or loss. In such photonic systems, the tran-sition from real to complex eigenvalues at the exceptionalpoint (EP) can be observed, depending on the interplaybetween the gain-loss and the coupling strength betweenthe eigenmodes of the subsystems. The operation regimenear the EP allows for the observation of intriguing phe-nomena, such as single-mode lasing [3,4], light trapping[5], and unidirectional invisibility [6,7].
Most of the photonic systems exhibiting EPs are realizedby balancing the gain and loss of the constituent mate-rials, which limits their practicality. To circumvent this
issue, EPs in optical systems with structurally broken PTsymmetry have been actively studied. For example, sys-tems with an asymmetric loss-only configuration, whichis a gauge-transformed PT-symmetric system, possess EPsforming at the transition between the states with complexeigenvalues [8]. Other works utilize radiation loss insteadof material loss [9], exploiting their mathematical equiva-lence in the optical Hamiltonian. Nevertheless, all opticalEPs that have been realized so far require energy exchangewith outer systems.
Here, we reveal the possibility of exhibiting an EP inoptical systems with neither energy gain nor loss, wherethe propagation constant is real at the EP. We arguethat, unlike many quantum-mechanical and optical systemswith energy as their eigenvalues, the Hermiticity of opti-cal Hamiltonians for guided modes in waveguides is notdirectly related to energy conservation, as their eigenvaluesare propagation constants of optical modes. As a represen-tative example, we investigate lossless I -M -I plasmonicwaveguides. Here, the I -M -I waveguide has eigenval-ues of its optical Hamiltonian transitioning from beingpurely real to becoming a pair of complex conjugates. Thisbehavior is strikingly similar to that of conventional PT-symmetric systems with balanced gain and loss [10,11],even though each eigenmode does not experience actualgain nor loss, while being a plasmonic mode without radia-tive loss. We also show that the EP in this system is robustto both P and T symmetry breaking. Furthermore, we
SUNG YOON MIN et al. PHYS. REV. APPLIED 14, 054041 (2020)
numerically demonstrate a third-order EP that is expectedto have strong potential in sensor applications [12,13] bycombining two EPs in the parametric space. Finally, weinvestigate asymmetric I -M -I structures with and withoutmaterial gain and loss, and we numerically demonstrate theconstant presence of the EP regime in all cases.
II. RESULTS AND DISCUSSION
A. Exceptional points in an I -M -I waveguide
We start by considering the general case of a two-levelsystem. The generalized mathematical condition to forman EP in a non-Hermitian two-level system with the Hamil-
tonian H =(
a bc d
)is given by [14] (a − d)2 + 4bc =
0. This condition allows an EP to be formed even with-out PT symmetry [15]: when Re[a] = Re[d], so that a − dis purely imaginary, while b = c∗, or when a − d is realand b = −c∗. In a system of coupled waveguides, a and dcorrespond to the propagation constants of the eigenmodesof the subsystems, while b and c correspond to their cou-pling coefficients [8,16]. For example, photonic systemswith symmetric real propagation constants and asymmetriclosses [8] or radiation modes [9], as well as PT-symmetricsystems, where [17,18] a = d∗, correspond to the case withpurely imaginary a − d and b = c∗. The condition b = c∗is generally satisfied in symmetric coupled waveguides[16], except for anti-PT-symmetric systems, where [19,20]a = −d∗ and b = −c∗. Therefore, most studies considerPT-symmetric systems with gain and/or loss, where a − d
(a)
(c)(b)
Een
rgy
velo
city
(10
–2 v
E/c
)
4
0
0.16Core thickness (ak0)
8
12
–4
EP
|f Ò
|bÒ
0.18 0.20 0.22 0.24
15
10
5
0
–5
|f Ò
|bÒ
0.16Core thickness (ak0)
0.18 0.20 0.22 0.24
Re Im
Effe
ctiv
e m
ode
inde
x n e
ff
FIG. 1. Exceptional point in the lossless I -M -I waveguide. (a)Schematic of the I -M -I plasmonic waveguide with Hz field pro-file of the two propagating eigenmodes with positive (|f 〉; blue)and negative (|b〉; red) energy velocity. (b) Energy velocity and(c) effective mode index of the eigenmodes in the lossless waveg-uide, showing the onset of the EP regime at αEPk0 ≈ 0.207;εI1 = εI2 = 12, εM = −10.
is purely imaginary. However, it is difficult to achievethe stable existence of an EP in conventional gain-lossphotonic systems because of the low efficiency, narrowbandwidth, and nonlinear instability of gain materials. Theselection of proper materials is also very limited. To avoidthese problems while accessing the EP regime, one mayconsider a plasmonic system with purely imaginary a andd, so that a − d is always purely imaginary. The sim-ple I -M -I plasmonic waveguide is a handy example ofsuch a system. Figure 1(a) schematically depicts the I -M -Iplasmonic waveguide comprised of the two I -M inter-faces, with the semi-infinite insulator layers I 1 and I 2 (withdielectric permittivities εI1,2; Re[εI1,2] > 0), and the metalcore M (Re[εM ] < 0) of thickness α. Then, we defineρj = |Re[εIj ]/Re[εM ]| and γj = Im[εIj ]/Re[εIj ]; j = 1,2.The plasmonic dispersion relation for a single I -M inter-face is given by βI -M = k0nI -M = ±k0
√εMεI/(εM + εI ),
where βI -M and nI -M are the propagation constant andeffective mode index of surface plasmon, respectively, andk0 = 2π/λ0 is the free-space wave number. The dispersionof the I -M -I waveguide is known to be [21,22]:(
1 + εMκI1
εI1κM
) (1 + εMκI2
εI2κM
)
=(
1 − εMκI1
εI1κM
) (1 − εMκI2
εI2κM
)e−2ακM , (1)
where κj = iky,j =√
β2 − εj k20 is the decay constant along
the y axis, α is the metal core thickness, and β is thepropagation constant of the I -M -I eigenmode in the xdirection.
When ρ1,2 = ρ > 1 and the system has neither gainnor loss (γ1,2 = 0), the I -M -I waveguide can supporttwo propagating TM0 eigenmodes with even parity of thegoverning Hz field (see Fig. 5 in Appendix) and antipar-allel energy velocity, while their phase velocity is of thesame sign [23]. The energy velocity is given by vE =∫
Sxdy/∫
udy, where Sx is the x component of the Poynt-ing vector, and u is the time-averaged energy density [24].For convenience, we call the mode with positive vE aforward mode, |f 〉, and that with negative vE a back-ward mode, |b〉. As the core thickness increases, vE ofboth eigenmodes converges to zero [Fig. 1(b)], and theireffective mode indices, neff = β/k0, become degenerate, asshown in Fig. 1(c). At this point, not only do the eigen-values coalesce, but also the field profiles (see Fig. 5 inAppendix), which is a pronounced characteristic of the EP.In the symmetric I -M -I waveguide (εI1 = εI2), the groupvelocity of each eigenmode ∂ω/∂β and the geometry-related parameter ∂α/∂β can be derived from dispersionEq. (1):
where c is the speed of light. Notably, the coalescence ofthe eigenmodes leads to the formation of an EP, where thegroup velocity ∂ω/∂β and the geometry-related parame-ter ∂α/∂β simultaneously vanish for both eigenmodes atthe critical core thickness αEP = 2κ−1
I [1 − (n2eff/n2
I -M )]−1.Interestingly, it has only been recently highlighted thatthe eigenmodes’ coalescence at exceptional points andthe vanishing group velocity are theoretically linked insystems with a certain distribution of gain and loss [5].
The manifestation of an EP in the I -M -I waveguide canbe understood by investigating the properties of the sys-
tem’s optical Hamiltonian H =(
a bc d
), where a and
d correspond to the eigenvalues of the two isolated I -Minterface modes. In the regime when ρ1,2 > 1 and γ1,2 = 0,each isolated I -M interface supports only a nonpropagat-ing leaky mode with purely imaginary βI -M [21,22], whichsatisfies the mathematical condition to form an EP, asa − d is purely imaginary.
In the I -M -I waveguide with a finite core thickness, thecoupling between the two I -M interfaces (b and c terms inthe optical Hamiltonian) results in a pair of eigenmodesthat are no longer leaky, but confined in the transversedirection. In weak coupling regime (α > αEP), the effectiveindices of eigenmodes are a pair of complex conjugates,corresponding to a system with balanced gain and loss,although the considered plasmonic modes do not experi-ence any actual energy gain or dissipation. As the metalliccore gets thinner, the coupling between the two I -M inter-faces grows and eventually balances out the “fictitious,” ormathematical, gain-loss at α = αEP, allowing for the twomodes to coalesce at the EP with a real effective index. Asthe core thickness further decreases (α < αEP), the cou-pling strength outweighs the fictitious gain-loss and theeffective indices of the propagating modes |f 〉 and |b〉divert from each other, as shown in Fig. 1(c), similar tothe case of a PT-symmetric system [10].
The behavior of |f 〉 and |b〉 modes in the I -M -I waveg-uide is nearly identical to that of the modes in coupledwaveguide systems with balanced gain and loss, except fortwo key differences. First, the energy flux (∫ Sxdy) of bothmodes converges to zero at the αEP, and the modes are non-propagating when α > αEP. Notably, in conventional gain-loss balanced systems, the eigenmodes still have nonzeroenergy velocities, even beyond the EP where the effectiveindices of the modes become complex. Consequently, wespeculate that the encircling of the EP in the parametricspace of the optical Hamiltonian, as exploited in many pre-vious studies [25–27], would not be possible in the losslessI -M -I configuration. Second, more importantly, the I -M -Iplasmonic waveguide with purely real permittivities can be
considered a closed system without energy exchange withthe environment, since both |f 〉 and |b〉 have zero radiationloss, as their fields decay exponentially further from theinterfaces (Fig. 5 in Appendix). The absence of radiativeloss is consistent with high effective indices of the eigen-modes, which cannot couple to the radiative modes withoutan additional phase-matching mechanism. We note thatspecial cases of I -M -I waveguides also support the low-index long-range surface plasmons that couple to radiativemodes [28]; however, this is not the case in this study.
At first glance, it may sound contradictory that a Hamil-tonian of a system without energy gain or loss could benon-Hermitian. However, we point out that the Hermiticityof optical Hamiltonians for guided modes in waveguidesdoes not require energy conservation, since the eigenvaluesare propagation constants of the optical modes [2]. Accord-ingly, the spatial coordinate along the axis of propagation(in our case, the x axis) in the paraxial equation plays therole of time t in the Schrodinger equation [10]. In typicaloptical systems, these differences are not pronounced, asthe phase and energy of the modes propagate in a similarmanner. In contrast, the phase and energy propagations inour system are completely different. In the weak couplingregime (α > αEP), the effective indices of the modes area pair of complex conjugates, whereas the energy veloc-ity vanishes. In the strong coupling regime (α < αEP), theeffective indices of the modes are real numbers with thesame sign, whereas the energy velocities of |f 〉 and |b〉have opposite signs. As a result, the behavior of the modeindices and the eigenvalues of the optical Hamiltonian aremathematically equivalent to that of a pair of waveguideswith balanced gain and loss, allowing an EP to be formed,even in a structure without energy gain or loss.
B. Exceptional points in a symmetric I -M -I waveguide
We proceed with analyzing the properties of the EP asmaterial loss and gain are introduced in a balanced manner(i.e., γ = ±γ1 = ∓γ2) to the symmetric I -M -I waveguidewith ρ1 = ρ2 = ρ > 1. Figures 2(a) and 2(b) show howneff and αEP vary, depending on the system parameters,respectively, when εM is fixed to −10. At a constant ρ, weobserve that EP occurs at smaller core thicknesses as |γ |increases. With a fixed |γ |, αEP also decreases with increas-ing ρ, akin to balancing out larger gain and loss from theindividual modes’ indices with stronger coupling in a PT-symmetric coupled waveguide system. When nonzero γ
is introduced, the energy velocities of the modes beyondthe EP (α > αEP) are no longer zero, but actually conveyenergy along the propagation direction, as in the conven-tional coupled waveguide system with balanced gain andloss.
On the other hand, when ρ is sufficiently smaller thanone and γ = 0, each I -M interface supports propagatingsurface plasmons with real βI -M . The I -M -I waveguide
054041-3
SUNG YOON MIN et al. PHYS. REV. APPLIED 14, 054041 (2020)
15
10
5
0
–5
(a)
(d)15
10
5
0
–5
(c)ρ
1.0 1.5 2.0 2.5
|γ|
0.10
0.20
(b)
0.8
0.6
0.4
0.2
0
ρ0.3 0.5 0.7 0.9
|γ|
TM1
0.16Core thickness (αk0)
0.18 0.20 0.22 0.24
0.26Core thickness (αk0)
0.28 0.30 0.32 0.34
Re Im γ = 0 |γ| = 0.25
Re Im γ = 0 |γ| = 0.25
0.30
0.4
0.3
0.2
0.1
0
0.25
0.15
0.05
0.350.40
1.0
1.2
0.10
0.20
0.300.25
0.15
0.05
0.35>0.4
Cor
e th
ickn
ess
at E
P (α
EPk 0
)C
ore
thic
knes
s at
EP
(αE
Pk 0
)
Effe
ctiv
e m
ode
inde
x n e
ffEf
fect
ive
mod
e in
dex n e
ff
FIG. 2. Analysis of EP regime in symmetric I -M -I waveg-uide with and without material gain and loss. (a) Effective modeindex of eigenmodes with and without material gain-loss γ =Im[εI ]/Re[εI ] when Re[εI ] = 12, εM = −10 (ρ > 1), showingthe constant presence of the EP at coalescence of the eigenmodeswith antiparallel energy velocity. (b) Core thickness correspond-ing to the onset of the EP as a function of γ and ρ when ρ > 1;black dots indicate system parameters in (a). (c) Effective modeindex of eigenmodes with and without γ when Re[εI ] = 8, εM =−10 (ρ < 1): one even (|f 〉; blue) and one odd (TM1; green)eigenmode is supported by the structure, unless nonzero γ isintroduced, forbidding the existence of EP in the lossless case.(d) Same as in (b), when ρ < 1, showing divergence of αEP whenγ → 0; black dots indicate system parameters in (c).
supports one even (TM0) and one odd (TM1) mode [Fig.2(c)], both having real propagation constants and positiveenergy velocity. These two eigenmodes cannot becomedegenerate because of different field parity. In this case,EP can be retrieved only with symmetrically introducedgain and loss [Fig. 2(c)], which would correspond to a con-ventional PT-symmetric system [10,11,29]. Therefore, αEPdiverges as γ → 0 [Fig. 2(d)], and EPs are exhibited onlyif |γ | > 0. Here, again, αEP decreases with increasing |γ |due to a stronger coupling being required to balance out theincreased gain-loss. The dependence of αEP on ρ may seemsomewhat nontrivial, but can be explained using similarreasoning of coupling balancing out gain and loss [10].
Interestingly, we find that the dispersion of losslessI -M -I waveguides possesses nontrivial fine structures fortransitional cases at ρ ≈ 1, as summarized in Fig. 3. Thepreviously discussed case withρ ≥ 1 (at γ = 0) is shownin Fig. 3(a). In Fig. 3(b), we plot the eigenvalues fromFig. 3(a) as a function of α on a complex plane to clearlyshow their dynamics near the EP (shown by the red dot)as α transitions through αEP. The arrows indicate theevolution of eigenmodes from propagating (solid arrows)
to nonpropagating (dashed arrows) as the core thicknessincreases. When ρ becomes slightly less than one, twoadditional modes of different parity with positive energyvelocity enter the solution space [Fig. 3(c)]. Among theTM0 modes, the backward mode, |b〉 (red), coalesces witheach of the two forward modes, |f1〉 and |f2〉 (blue), so thattwo separate EPs are formed in the system, as demon-strated in Fig. 3(d) on a complex plane. As we furtherdecrease ρ, the two separate EPs merge at the same αEP, asshown in Fig. 3(e). Then, three TM0 eigenmodes coalesceat a single αEP, forming a third-order EP [encircled red dotin Fig. 3(f)]. The value of ρ associated with a third-orderEP made from the fundamental modes is the minimumpossible for a given system, and no EPs are present if ρ
is smaller than this value, unless actual gain and loss areintroduced into the system, as shown in Figs. 3(g) and 3(h).
We note that, in a conventional photonic system, threesubsystems with propagating modes are required to forma third-order EP. The excitation and simultaneous coales-cence of three eigenmodes is not a trivial task, and hence,complicated geometries are suggested to form a third-orderEP [13,30]. Here, however, we show that a third-order EPcan be formed, even in a simple I -M -I waveguide.
The intersection angle between the eigenvalue traceson the complex plane represents the degeneracy order atthe EP. If we assume a system with n eigenmodes, it canbe described by an n × n Hamiltonian. In the vicinity ofthe nth order EP, where the Hamiltonian solutions canbe approximated as straight lines [Figs. 3(b), 3(d), 3(f)and 3(h)], the Hamiltonian can be written as an nth orderequation with n degenerate roots near the EP, as follows[31]:
(λm − βEP)n = rneinθm = ξ , (4)
where λm are the eigenvalues of the Hamiltonian, βEPis the degenerate eigenvalue at the EP, Re(ξ) = rn, andθm = 2πm/n (m = 0, 1, 2, · · · , n − 1). Then, the eigen-values of the Hamiltonian are given by λm = βEP + reiθm ,and thus, the parameter ξ determines whether the eigen-values are real or complex and is zero at the EP. Fromthe perspective of the coupled mode theory in a conven-tional PT-symmetric structure (n = 2), for example, ξ =|κc|2 − |Im(βsub)|2, where κc is the coupling coefficientand βsub is the propagation constant of an eigenmode ofthe subsystem. In the real eigenvalue regime, the value ofξ is positive, and thus, all eigenvalue traces intersect atthe EP at a constant angle of ϕn = π/n = θm/2m. In thecomplex eigenvalue regime (or PT-broken regime), ξ isnegative, which can be treated as a positive real value withθm = (2m + 1)π/n. Therefore, upon the transition throughthe EP, the eigenvalue traces intersect at the EP at a con-stant angle of ϕn = π/n = θm/2m, as we show in Figs.3(b), 3(d) and 3(f).
FIG. 3. Formation of third-order exceptional point in lossless symmetric I -M -I waveguide. Top row: effective indices of the eigen-modes as a function of core thickness for systems with different ρ (εM = −10): (a) εI = 10, (c) εI = 9.7, (e) εI ≈ 9.65, and (g)εI = 9.6. Bottom row: same effective indices plotted on a complex plane. (b),(d) Second-order EPs (red dots) corresponding to (a),(c), respectively. (f) Coalescence dynamics of three eigenmodes shown in (e) forming the third-order EP. No EPs exist for smaller ρ,as demonstrated in (g) and (h). Arrows indicate evolution of the effective index of propagating (solid) and nonpropagating (dashed)eigenmodes as core thickness increases. Intersection angle ϕn = π/n corresponds to degeneracy order n.
C. Exceptional points in an asymmetric I -M -Iwaveguide
Finally, we consider I -M -I structures with broken Psymmetry (εI1 = εI2). Since a − d stays purely imaginaryif ρ1,2 > 1 and γ1,2 = 0, P symmetry is dispensable formanifestation of EPs in asymmetric lossless I -M -I waveg-uides. To illustrate the constant presence of the EP regime,we calculate αEP for the asymmetric I -M -I waveguide
as a function of ρ1 and ρ2, as shown in Fig. 4(a), whilemaintaining εM = −10. Similar to the previous cases, αEPdecreases as either ρ1 or ρ2 increases. In contrast, thesystem with broken P symmetry and γ1,2 = 0 requires aspecific ratio of gain and loss of ±γ2,EP/∓γ1,EP to accessthe EP regime, depending on the values of ρ1,2. We cal-culate the required ratio of −γ2,EP/γ1,EP as a function ofρ1,2, confirming the consistent presence of EPs [Fig. 4(b)].
r 2
r1
–g2,
ER/g 1,
ER
Re(neff)6.0 7.0
0.4
0.2
0.0
–0.2
Im(n
eff)
–0.4
(b) (c)
r1
1.0 1.5 2.0 2.5
r 2
1.0
1.5
2.0
2.5 0.30
0.25
0.20
0.15
0.10
(a)
j2r 1 = r 2
6.2 6.4 6.6 6.8
00.10.25
-g1
2
1
1.8
1.6
1.4
1.2
1.0 1.2 1.4 1.6
4
0.5
0.251.8
1.0
Cor
e th
ickn
ess
at E
P (
a EPk
0)
FIG. 4. Analysis of the exceptional point regime in asymmetric I -M -I waveguide with and without material gain and loss. (a)Core thickness αEPk0 corresponding to the EP regime, when ρ1,2 > 1 and γj = 0. (b) Specific ratio between material gain-loss intwo insulator layers (−γ2,EP/γ1,EP) required for the manifestation of EP as a function of ρ1,2. (c) Complex plane plot of effectiveindices of waveguide eigenmodes, showing second-order EPs (red dots) formed when γj = 0 (black lines) andγj = 0 (red and bluelines); Re[εI1]= 12, Re[εI2] = 11, and εM = −10, as indicated by the black dot in (a) and (b). Arrows indicate evolution of theeigenmodes’ effective indices before (solid) and after (dashed) onset of the EP regime as core thickness increases.
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SUNG YOON MIN et al. PHYS. REV. APPLIED 14, 054041 (2020)
We note that the condition ρ1,2 > 1 must be satisfied; oth-erwise, EPs cannot be formed, even if gain and loss areintroduced.
The complex-plane trace of the eigenvalues in the asym-metric structure near the EP regime is shown in Fig. 4(c).Interestingly, the eigenvalues of the system with both Pand T symmetries broken are not required to be real tomanifest the EP in the complex eigenvalue regime. Theunbalanced gain and loss are reflected as the rotation ofthe eigenvalue traces near the EP, while the intersectionangle ϕ2 = π/2 is preserved. We note that this type of EPhas been widely studied in other photonic systems, includ-ing passive PT symmetry with asymmetric material [8] orradiative losses [9].
III. CONCLUSIONS
We demonstrate the emergence of EPs in an I -M -I plas-monic waveguide possessing neither gain nor loss, includ-ing the case when P symmetry is broken. The subtle dis-tinction between the Schrodinger equation and the opticalparaxial equation for waveguides allows an optical Hamil-tonian to be non-Hermitian without energy exchange withthe environment, when materials with negative permittiv-ities are introduced to the system, and thus, the directionsof energy and phase propagation are no longer parallel.Furthermore, we show the possibility of combining EPsto generate a higher-order EP in a simple plasmonic struc-ture. Generally, the necessity of simultaneously excitingthree eigenmodes that coalesce in the EP regime to accesshigher-order EPs in photonic systems limit the practicality.Our study suggests that the generation of a third-order EPis possible in a simple I -M -I structure, which relaxes thegeometrical and material restrictions and gives an insightinto generating higher-order EPs. Finally, we analyze caseswith gain and loss in the system, showing that achievingan EP in the system with broken P symmetry requires anasymmetric gain and loss profile. We speculate that ourfindings are not limited only to plasmonic systems, but alsoextend to a wider class of photonic systems with decoupledphase and energy propagation.
ACKNOWLEDGMENTS
This work was supported by the National ResearchFoundation of Korea (NRF) funded by Ministry of Sci-ence and ICT (Grants No. 2017R1E1A1A01074323 andNo. 2016M3D1A1900038). S.Y.M. and S.G.M. acknowl-edge support from the NRF Young Researchers pro-gram funded by Ministry of Science and ICT (Grant No.2019R1C1C1011131).
APPENDIX: MAGNETIC FIELD PROFILE OF THEEIGENMODES IN AN I -M -I WAVEGUIDE
When α < αEP, the waveguide supports two propagat-ing TM0 eigenmodes with one |f 〉 and one |b〉 mode (see
1.2
1.0
0.0
yk0
–1.5 –1.0 –0.5 0.0 0.5
ak0 = 0.16
(a)
(b)
Mag
netic
fiel
d H
z (no
rmal
ized
)
1.0 1.5
0.8
0.6
0.4
0.2
Mag
netic
fiel
d H
z (no
rmal
ized
)
yk0
–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5
1.0
0.8
0.6
0.4
0.2
0.0
–0.2
–0.4
Im{Hz}
aEPk0 = 0.207
ak0 = 0.25
Re{Hz}
Re{Hz}
FIG. 5. Magnetic field profile of TM eigenmodes in losslessI -M -I waveguide. (a) Purely real magnetic field of propagat-ing eigenmodes before onset of the EP regime (blue, forwardmode; red, backward mode) and coalescent field profiles of bothmodes at EP (black). (b) Real and imaginary parts of the mag-netic field for nonpropagating eigenmodes after onset of the EPregime. εI = 12, εM = −10 and dashed lines indicate location ofI -M interfaces.
Fig. 1(c) for details). In terms of the governing mag-netic field Hz, the eigenmodes have nonzero Re[Hz], whilelm[Hz] = 0, as shown in Fig. 5(a). As the core thick-ness increases, the two eigenmodes (Fig. 5(a), blue andred) coalesce at the EP when α = αEP (Fig. 5(a), black),still having a real effective index. When α > αEP, the twoeigenmodes split again, with their effective indices beinga pair of complex conjugates, which is reflected in thefield profile (Fig. 5(b)). This demonstrates that the systemalways supports two eigenmodes except for the EP.
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