Qionggan Zhu(朱瓊干), Lichen Chai(柴立臣), and Hai Lu(路海)

1Department of Science,Taiyuan Institute of Technology,Taiyuan 030008,China

2Engineering Laboratory for Optoelectronic Technology and Advanced Manufacturing,School of Physics,Henan Normal University,Xinxiang 453007,China

Keywords: zero-index metamaterial,non-Hermitian,gain-loss interaction,magnetic flux

1.Introduction

Nowdays non-Hermitian optic prevails among researchers in related fields due to their hyperphysical applications such as communications and lasing.[1-3]The PT symmetry can be implemented in optical systems because the gain and loss are easy to realize experimentally.By introducing equal amounts of gain and loss into a material to ensure that the refractive index satisfiesn(x)=n?(-x), a PT-symmetric optical system is formed.In such a system, two or more modes can merge and undergo a phase transition from real energy spectrum to complex energy spectrum at exceptional point (EP).[4]The PT symmetry not only provides additional spatial modulation in optical systems, but also brings about new physical effects,such as negative refraction,[5]asymmetric invisibility,[6]and splitting of bound states in the continuum(BICs)into two states with purely real energy: PT-BICs and threshold states.[7]

Zero-index metamaterial (ZIM) refers to material whose dielectric constant and magnetic permeability are both close to zero or one of them approaches to zero,i.e.epsilon-nearzero(ENZ)and magnetic permeability-near-zero(MNZ)materials.In the past decade, ZIM have been widely used in energy squeezing,[8]wavefront manipulation,[9,10]waveguide bending,[11]energy flow control,[12]full transmission and reflection and optical field enhancement[13,14]because of their quasi-static field distribution.In addition,waveguide systems based on ZIM can achieve BICs without geometric symmetry requirements, providing new ideas for studying BICs in disordered systems.[15]Usually,the realization of MNZ materials is difficult and their application frequency is low,making their application range extremely limited.In recent years,researchers have shifted their focus to utilizing ENZ materials to realize effective material parameters for ZIM systems,from the perspective of material implementation and broader application frequency range.[16,17]Using the concept of photon doping, ENZ media can achieve effective magnetic response in a wide range of parameter regions.It means that the effective material parameters of ZIM systems can be customized by doping macroscopic media objects.Regardless of the position,size and number of doping objects,the composite material structure is equivalent to a uniform system with constant refractive index distribution.[18,19]

Since Viet Cuong Nguyenet al.[20]realized full transmission and reflection of light based on ZIM waveguide loaded with defect structures more than a decade ago, many unusual optical phenomena have been generated on this platform based on ZIM’s quasi-static and photon-doping effects.[21,22]In previous studies,the system was generally considered to be ideal with no losses, and gain was later introduced to compensate for unavoidable losses.In fact, the interplay between gain and loss may have far-reaching implications that extend well beyond the mere compensation.The concept of quantum PT symmetry has inspired new ideas.Non-Hermitian optics extends conventional real-parameter space modulation to the complex space, where losses are not necessarily regarded as harmful effects.In some cases, they facilitate the interaction between light and matter to produce abnormal phenomena.In the SPP waveguide bypass coupled cavity structure,the competing mechanism between loss and gain produces some counterintuitive effects.[23]The combination of ZIM’s quasi-static characteristics and PT symmetry can also produce rich physical effects,opening up new ideas for light field control, such as PT-induced asymmetric transmission,[24]CPA laser modes,[25]PT symmetry suppression,[26]and perfect resonance transmission.[27]

To investigate the intricate mechanisms of interactions between loss and gain in complex systems more comprehensively, this paper focuses on a two-dimensional (2D) ZIM waveguide doped with two gain and loss defetcts as the research platform.The responses of the system under non-Hermitian regulation of loss and gain are investigated under two conditions: ZIM filling media are epsilon-and-mu-nearzero (EMNZ) and epsilon-near-zero (ENZ).Moreover, the physical mechanism of abnormal transmission characteristics of waveguide is explored from the perspective of the magnetic flux and fields locality.The new interaction rules between loss and gain under the background field are revealed to provide some theoretical guidance for the application of non-Hermitian optical physics.

2.Model building

The schematic diagram in Fig.1 illustrates the structure of a 2D waveguide filled with a ZIM medium, featuring two dielectric cylinder defects doped within the ZIM.The upper defect embedded in the ZIM meduim is the lossy cavity and the below one is the gain cavity.The permittivities of media in two chambers are assumed to beεd-iγlandεd+iγgrespectively.In general, the loss can be easily obtained by nature materials, but not easy for gain medium.The fitted gain can be induced by laser systems or nonlinear multiple-wave mixing through external pumping.[28,29]Additionally,we consider an electromagnetic(EM)wave withHinc=Hincexp(ik0x)incident from the left inside the waveguide, wherek0=ω/cis the wave vector andω=2π fis the angular frequency of incident wave.The working frequencyfis set to 15 GHz in our calculation.TheHincfield is polarized in thezdirection,and theEincfield is directed along theyaxis.

Fig.1.Schematic diagram of the 2D air-ZIM-air waveguide loading gain and lossy cavities structure.The upper and lower boundaries are PECs.The width and length of the ZIM junction are w and d,their values are w=60 mm and d=44 mm,respectively.The radii of the gain and lossy cylinders are R1 and R2 (R1=R2=8 mm).

3.Theoretical derivation

To solve the transmission characteristics of the ZIM waveguide loaded defects,we can refer to Ref.[21].By utilizing Maxwell’s equation, the following Helmholtz equation is derived:

whereHi(i=1,2)is the local magnetic field in the defect objects andH0is the quasi-static field of the ZIM background medium.The magnetic field and electric field inside each cylinder are obtained by the differential equation using the separation variable method.Then, by applying the Faraday-Maxwell equation and Stokes’theorem,we can further calculate transmission coefficient

whereEis the electric field in the ZIM region andEiis the electric field in each defect.Integral sign subscript?Cis the boundary enclosing the whole ZIM region, and?Cidenotes the boundary of each defect, The transmittance coefficienttof the ZIM waveguide loading two-defect structure can be described as[15]

whereAi(i=1, 2)is the magnetic potential at the boundary of the two defective bodies.The magnetic flux of the defect cavityφiis written as

where J0and J1are the zero-order and first-order Bessel functions of the first kind respectively,is the wave vector in the objects,cis the speed of light in vacuum.

3.1.Case I

When the two defect bodies are doped in the epsilon-andmu-near-zero ZIM medium (ε ≈0,μ ≈0), they can be regarded as two resonant chambers.The propagation properties of waveguide are independent of the defects’ location in the ZIM.When two cavities introduce loss and gain respectively,the transmission coefficient can be written as

whereφlandφgrepresents the magnetic flux of the loss cavity and the gain cavity respectively and they can be obtained by Eq.(4).We can observe that the transmission coefficient of the system is determined by the magnetic flux of both the loss and gain chambers.

3.2.Case II

When dealing with an ENZ-based ZIM host(ε ≈0,μ=1), the transmission coefficient of the 2D waveguide with the lossy and gain dopants can be represented as follows:

where the termφ0=μH0(s-sd)/crepresents the magnetic flux of the quasi-static fieldH0through the ZIM region of waveguide,sdenotes the entire area of ZIM and the defects area andsdrepresents the area of two defects within the cavity.From Eq.(6),the transmittance coefficient of the waveguide is determined by the sum of the magnetic fluxes of both the loss and gain chambers plus ZIM.Therefore,the ENZ-based ZIM medium has characteristic of magnetic response due to nonzero magnetic flux,which provides a background field for the interaction of the loss cavity with the gain cavity.

4.Result and discussion

4.1.Case I

For the epsilon-and-mu-near-zero ZIM filling medium,due to the quasi-static field effect of ZIM, two defect bodies doped in the waveguide can be considered as two mutuallycoupled, location-free resonant chambers.[15]When the defects are ideal and lossless,the mutual interference of the two chambers generates a resonance mode I,as shown in Fig.2.

Fig.2.The 2D map of the transmission coefficient as a function of the permittivity εd and the asymmetry parameter α.An ideal BICs occurs at α =0 and εd=4.82(the white dotted circle).

The size difference between the two cavities radii is expressed by the asymmetry parameterα, whereα= (R2-R1)/R1.As the parameterαchanges from nonzero to zero,mode I undergoes a transition from resonance mode to bound states in the continuum(BICs).If two resonant chambers introduce an equal amount of loss and gain separately(γl=γg),as shown in Fig.3(a),the dissipation of the loss cavity and the amplification of the gain cavity are balanced.A transmission peak and a reflection dip appear aroundεd=4.82, and the sum of reflectance and transmission equals one.Therefore,the system remains generally balanced.However, when the loss and gain are not equal (γl/=γg), the mutual competition between loss and gain leads to some counter-intuitive transmission characteristics in the waveguide.i.e.whenγl＞γg,as shown in Fig.3(b),the sum of the reflectance and transmission are greater than one aroundεd=4.82,EM waves are cumulative and the system behaves as an amplifier.Conversely,whenγl＜γg,EM waves are dissipated and the system behaves as an attenuator.

Fig.3.The transmission coefficient t and reflection coefficient r as well as the sum of reflectivity R and transmittance T vary with the dielectric constant εd.(a) The case for the loss is equal to the gain(γl =γg =0.02).(b) The sum of reflectivity R and transmittance T varies with the dielectric constant εd for the loss unequal to the gain.The permittivity and permeability of the ZIM host are ε =μ =10-4.

The underlying physics of anomalous transmission of the system is revealed from the perspective of magnetic flux and field locality.Due to the introduction of loss and gain,the magnetic fluxes of defects become complex.Forγl＞γg, as depicted in Fig.4(a), we have Re(φg)＞Re(φl) and-Im(φg)＞Im(φl)near the resonant transmission peak.Conversely,Re(φg)＜Re(φl)and-Im(φg)＜Im(φl)are observed in Fig.4(b)forγl＜γg.It shows that the larger coefficient of gain/loss cavity, the smaller real and imaginary parts of the magnetic flux.The difference of magnetic flux between two cylinders with gain and loss plays a crucial role in the counterintuitive electromagnetic transmission through ZIMs waveguide.In which the cavity has the largest magnetic flux, and it dominates.Therefore, for the larger loss case, the magnitude of the complex-form magnetic flux of the gain cavity is greater than that of the lossy cavity(|φg|＞|φl|),as illustrated in Fig.5(a).It is shown that the stronger field is localized in the gain cavity compared to that in the loss cavity.Consequently,the gain cavity plays a dominant role,and the system exhibits an amplified response.However, once the gain is greater, as shown in Fig.5(b),the magnitude of the magnetic flux of the gain cavity translates to less than that of the loss cavity near the transmission peak (|φg|＜|φl|), and the local field of the loss cavity becomes stronger.As a consequence,the loss cavity takes over,and the system exhibits a dissipated response.

Fig.4.The analytical φl and φg as a function of the permittivity εd.(a)Real part(solid red line)and imaginary part(dashed red line)of φg and real part(solid blue line)and imaginary part(dashed blue line)of φl for γl=0.04,γg=0.03.(b)Real part and imaginary part of φg and φl for γl=0.03,γg=0.04.

Fig.5.Numerically calculated modulus of magnetic flux|φ|varies with the dielectric constant εd for the EMNZ medium(ε ≈0,μ≈0).(a)The case for the larger loss(γl =0.04,γg =0.03).(b)The case for the larger gain(γl =0.03,γg =0.04).The simulated magnetic field modulus distribution in the ZIM region is shown respectively in the inset of panels (a) and (b).Numerical simulation is performed by using the finite element method software COMSOL Multiphysics.The numerical calculation of the magnetic flux modulus is obtained by the definition, that is, the integral of the magnetic field over the surface area(|φi|=|∫s Bi·ds|).

4.2.Case II

For the ENZ-based host(μ/=0)case,as shown in Fig.6,the system supports two radiative monopole modes, and the mechanisms of the two modes are different.Mode I is similar to the mode I pattern in the case I, and it originates from the direct interference of the two chambers.As the asymmetric parameterαchanges,there is also a conversion from BICs to resonance mode.Mode II is a Fano resonance which is based on the effect of the ZIM quasi-static field.It can be inferred from Eq.(6),magnetic fieldH0of the ZIM medium has a nonzero magnetic fluxφ0, hence mode II is produced by the two cavities seeing a resonant structure that interferes with the continuous quasi-static fieldH0in the medium.As the parameterαchanges,resonance linewidth of mode II has no significant change.

Fig.6.The 2D map of the transmission coefficient as a function of the permittivity εd and the asymmetry parameter α.

When the equal loss and gain are introduced,as shown in Fig.7,two transmission peaks appear in the reflection valleys aroundεd=4.82 andεd=4.915,respectively.The linewidth of mode I is narrower than that of mode II.The sum of reflectance and transmission is equal to one.Under asymmetric control of loss and gain, the eccentric and different transport effects of waveguide corresponds to mode I and mode II.

Fig.7.The transmission coefficient t and reflection coefficient r as well as the sum of reflectivity R and transmittance T vary with the dielectric constant εd for the loss being equal to the gain (γl =γg =0.02).The permittivity and permeability of the ZIM medium are ε =10-4 andμ =1.

Fig.8.The sum of reflectivity R and transmittance T varies with the dielectric constant εd.The solid pink line plus the triangle mark represents the theoretical calculation,and the blue-green solid line plus the triangle mark represents the simulation result.(a) The case for the loss is greater than the gain(γl=0.04,γg=0.03).(b)The case for the loss is less than the gain(γl=0.03,γg=0.04).

Whenγl＞γg,as shown in Fig.8(a),for mode I,the sum of reflectivity and reflectivity is less than 1,the system displays dissipative,while for mode II,the situation is just the opposite,the sum of transmittance and reflectivity is greater than 1,the system turns to be cumulative.Whenγl＜γg,it can be seen in Fig.8(b),for mode I,the sum of reflectivity and reflectivity is greater than 1,the system is cumulative,and for mode II,the sum of transmittance and reflectivity is less than 1,the system becomes dissipative.

Similarly,the mechanism behind it can be analyzed from the magnetic flux and local cavity field.For the ENZ case,the real and imaginary values of the magnetic fluxφlandφgare identical in case I according to Eq.(4).Magnetic flux plotting is thus not repeated for the ENZ case.However,the analytical expression Eq.(4)presents the magnetic flux of individual defect,but it fails to describe the magnetic flux while the defects interact with the background field.The magnetic flux modulus of each cylinder was analyzed to further reveal the physics,with the simulated results shown in Fig.9.

Fig.9.Numerically calculated of the magnetic flux|φ|varies with the dielectric constant εd for the ENZ medium(ε ≈0,μ=1).(a)The case for the larger loss(γl=0.04,γg=0.03).(b)The case for the larger gain(γl =0.03, γg =0.04).Each inset in the graphs shows the simulated magnetic field modulus in the ZIM region at the resonant peak.

Whenγl＞γg, for mode I, it can be seen from Fig.9(a)that the magnetic flux and field local strength of the gain cavity are much larger than that of the lossy cavity,the amplification of the gain cavity is enhanced by the stronger field localization, therefore the gain cavity dominates, and the system is accumulated.The interaction between the loss cavity and the gain cavity is competitive for mode I.For mode II, the magnetic flux and local field strength in the gain cavity are close to that in the loss cavity, which is not enough to enhance the amplification of the gain cavity.The loss and gain cavities compensate each other.Because the loss cavity coefficient is larger,the overall performance of two cavities is lossy,and the system is dissipated.Whenγl＜γg,as shown in Fig.9(b),for mode I,the magnetic flux and field strength in the loss cavity are much larger than those in the gain cavity,which enhances the lossy effect of the loss cavity.The loss cavity thus dominates and the system is dissipated.For mode II,the magnetic flux of the loss cavity and the localized field strength are a little larger than those of the gain cavity, the effect of the total mutual compensation of the two chambers is the gain, so the system is cumulative.

5.Conclusion

The concept of photonic doping can be extended to non-Hermitian scenarios,where gain and loss are distributed in two dop bodies.The non-Hermitian interaction between loss and gain in complex systems is studied by using a 2D ZIM waveguide with loaded loss and gain defects as the platform.For the case of EMNZ-based ZIM medium,our results show that EM waves are cumulative and the system behaves as an amplifier under the resonant mode when the loss cavity coefficient is greater than the gain cavity coefficient.Conversely,when loss less than gain, EM waves are dissipated and the system behaves as an attenuator.For the case of ENZ host medium,the system supports two resonant modes based on the quasi-static field effect of ZIM.By tailoring the gain and loss, it is found that the responses of the system in these two modes are completely different.When the loss coefficient surpasses the gain,the system response becomes dissipative under one mode and accumulative under the other mode,and vice versa.The physical mechanism of the system’s anomalies is explained from the perspective of magnetic flux and field locality.Our paper provides theoretical guidance for the application of light amplification and the gain-loss interaction of matter.

Acknowledgement

Project supported by Scientific and Technological Innovation Program of Higher Education Institutions in Shanxi Province,China(Grant No.2021L554).