Enhancing the Goos–H¨anchen shift based on quasi-bound states in the continuum through material asymmetric dielectric compound gratings

Xiaowei Jiang(江孝伟), Bin Fang(方彬), and Chunlian Zhan(占春连)

1College of Optical and Electronic Technology,China Jiliang University,Hangzhou 310018,China

2College of Information Engineering,Quzhou College of Technology,Quzhou 324000,China

Keywords: bound state in the continuum,Goos-H¨anchen shift,dielectric compound grating,material asymmetry

1.Introduction

The interaction between light and matter has always been the core of nanophotonics research.[1,2]Realizing light localization in subwavelength structures with ultra-long radiation lifetimes is crucial for applications such as lasers, modulators,nonlinear optics and quantum computing.[3-7]Therefore,the bound states in the continuum (BICs) with infinite quality(Q)factor and superior light confining ability have aroused great interest.BIC is a ubiquitous physical phenomenon that was originally proposed in hypothetical quantum structures by Neumann and Wigner.[8]BIC is a local state that coexists with the continuum spectrum,is found in many physical structures,such as gratings or metasurfaces,[9-12]and has been used to realize interesting physical phenomena such as highQnear perfect absorption resonance with unpolarized focused light[13]and Goos-H¨anchen (GH) shift enhancement.[14,15]However,BIC is a dark mode with an invisible scattering spectrum,[16]which makes it impossible to detect in practice.By breaking the symmetry of the metasurface or grating to transform the BIC into a symmetry-protected quasi-BIC(QuasiBIC,QBIC)with a finite but extremely highQfactor and an ultra-narrow linewidth,[17,18]QBIC can be detected in the form of Fano resonance.

Two common methods to break the plane symmetry of a grating or metasurface are to change the geometry of the grating or metasurface unit[17-19]and to change the refractive index of the grating or metasurface unit material.[20-23]Most of the current studies are based on geometric asymmetry to achieve symmetry-protected QBIC resonance, but this approach suffers from the problem of imposing fundamental limitations on achieving the lowest geometric asymmetry,especially in the near-infrared and visible spectral bands.[23]TheQfactor is directly proportional to the negative quadratic of the asymmetry of the structure.[17-19]Therefore, theQfactor will significantly vary with the change in structural asymmetry,but there may be errors in the fabrication process,resulting in poor controllability.[19]Compared with geometric asymmetries, material asymmetry has the advantage of abundant means of implementation, such as by current injection,[24]photothermal effect,[25]Kerr effect[26]and epitaxy.[22]In addition, the material asymmetric grating or metasurface has strong controllability.Currently, there is not much research on material asymmetry in metasurfaces or gratings.Thus,we believe that the field requires further exploration in structural design and physical analysis.

Since it was first theoretically and experimentally demonstrated,[27]the GH effect has attracted attention as one of the important physical optics phenomena due to its profound physical significance and broad application scenarios.[28,29]The GH effect shows that when a beam of light is incident at the interface of two media and is totally reflected, the reflected beam will be laterally shifted relative to the path predicted by geometrical optics.[30,31]This lateral shift is called the GH shift.According to the stationary phase method,[32]the GH shift is proportional to the partial derivative of the reflection phase to the incident angle.Therefore,the GH shift can be enhanced through highQfactor resonance because the reflection phase will dramatically vary around the resonant angle.[14,15]Currently, different methods have been proposed to achieve highQfactor resonance,all of which significantly improve the GH shift,but their maximum GH shift is located at the reflectance dip.[33-36]In these studies,the reflectivity that corresponds to the maximum GH shift is even less than 0.1%,which is not conducive to the detection of GH shift signals in practical applications.

In recent years,it has been found that by using QBIC resonance,the GH shift can be enhanced,and the maximum GH shift can be guaranteed to be located at the reflection peak with perfect reflection.Zhang,[37]Wu[14,15]and Huang[38]used QBIC resonance to significantly enhance the GH shift in the near-infrared and terahertz bands, and the maximum GH shift was located at the reflection peak with unity reflectance.The QBIC resonance to enhance the GH shift in the aforementioned studies was achieved through geometric asymmetry.There are few studies based on material asymmetry to achieve QBIC resonance and even fewer studies based on material asymmetry to achieve GH shift enhancement.Although Berte,[20]Liu,[21]Yu,[22]Chen[23]and some others realized symmetry-protected QBIC resonance based on material asymmetry,none of them applied QBIC in GH shift enhancement.

In this paper, we propose to achieve QBIC resonance through dielectric compound grating with material asymmetry and enhance the GH shift by QBIC with the maximum GH shift at the reflectance peak.The effect of the grating structural parameters and material asymmetry on the QBIC resonance wavelength and linewidth is analyzed in detail.Theoretical calculations show that the GH shift can reach-980 times the resonance wavelength, and the effect of the grating structural parameters on the GH shift is investigated.The sensitivity of the grating as a refractive index sensor is calculated to be 1.96×106µm/refractive index unit(RIU).This study can provide a theoretical basis for the design and fabrication of GH shift tunable metasurfaces or gratings in the future.

2.Structural design and calculation methods

Figure 1(a) shows a three-dimensional schematic of the dielectric compound grating.In the figure,the dielectric grating consists of a silicon dioxide (refractive indexns=1.45)substrate, a silicon (refractive indexnsi= 3.45) waveguide layer and a compound grating layer.Within a grating periodP,there are two gratings with different refractive indices,one composed of silicon and the other composed of a material with refractive indexn.The material asymmetry parameter isγ=n-nsi.When an incident light(electric field linearly polarized along they-axis direction)is obliquely incident to the dielectric grating along thez-axis at an angle of incidenceθ,it is shifted by a distance along thex-axis direction(GH shift)and reflected to the free space at an angle of reflectionθ.In Fig.1(a), the GH shift can be divided into positive and negative GH shifts.Because the material asymmetry can excite the reflection QBIC resonance,it causes dramatic variation in the reflection phase around incident angleθ,which enhances the GH shift.Figure 1(b) shows specific parameters of the dielectric compound grating.The dielectric grating substrate thickness ishs= 220 nm, the waveguide layer thickness ishw=260 nm, the grating thickness ishg=160 nm and the width and spacing of the two gratings in a single unit arewandl=400 nm, respectively.Based on the above structural parameters, the QBIC resonance wavelength of the dielectric compound grating is around 1330 nm,this wavelength is useful in fields such as optical fiber communication and infrared night vision.Therefore, the dielectric compound grating that can achieve GH shift enhancement would be suitable for use as an optical device in optical fiber communication, infrared night vision,gas sensing and other related fields.

Fig.1.Schematic illustration of the dielectric compound grating.(a) Three-dimensional schematic of the dielectric compound grating;(b)cross-section of the dielectric compound grating.

The proposed asymmetric dielectric compound gratings in this paper can be fabricated using existing semiconductor fabrication technology.[19,21]First, a layer of silicon is deposited on the silica substrate by plasma-enhanced chemical vapor deposition(PECVD),and the required silicon grating is obtained through electron beam lithography (EBL) and reactive ion etching (RIE).Second, a dielectric layer with refractive indexnis deposited via epitaxial technique, followed by EBL and RIE,and a grating with refractive indexncan be obtained.Finally,the protective layer on the grating is removed by wet-etching to obtain the desired material asymmetric dielectric compound grating.

The permittivity of the grating layer can be expressed as[15,39]

whereεmis the grating Fourier harmonics andmis the order of the Fourier series.The grating Fourier harmonics can be expressed as

wherenair=1.From Eqs.(1)-(3), the choice of refractive indexnof the dielectric compound grating and the variation of the grating parameters will affect the dielectric compound grating permittivity.

The numerical theoretical calculation method in this paper is finite-difference time-domain(FDTD).To facilitate the calculation, we consider that the grating length along theyaxis is sufficiently long so that this paper calculates the twodimensional dielectric compound grating based on FDTD.Because the incident light is obliquely incident to the dielectric grating at an angle of incidenceθ,Bloch boundary conditions are added in thex-axis direction, while the perfect matchinglayer boundary conditions are added in thez-axis direction.

3.Results and discussion

3.1.Symmetry-protected QBIC resonance excited by material asymmetry

To demonstrate that the symmetry-protected QBIC resonance can be excited by the material asymmetry, we calculated the reflection spectra of dielectric compound gratings with different material asymmetry parametersγ, as shown in Fig.2.Currently,the incident angleθ=6°,w=200 nm andP=1200 nm.Figure 2(a) shows that whenγ=0, there is no resonance in the reflection spectra, but whenγ ̸=0, there is a clear Fano resonance.Asγgradually moves away from 0, the resonance wavelength will blueshift(γ ＜0)or redshift(γ ＞0), and the resonance linewidth gradually widens.The reason is thatγaffects the effective refractive index of the dielectric compound grating, and the resonance wavelength is related to the grating effective refractive index, as shown in Eq.(6).To make more intuitive observations, we show the resonance spectra in Fig.2(b), whereγis-0.05,-0.15 and-0.25.As shown in Fig.2(b),all cases achieve perfect reflection,the resonance linewidth widens and the resonance wavelength shows a blueshift whenγdecreases.This phenomenon and the references[9-11,17,40]can prove that the resonance atγ ̸=0 is a QBIC resonance and is derived from the symmetryprotected BIC transformation.

In Fig.2(a), identical absolute values of±γcorrespond to different grating resonance reflection spectra.To better understand the demonstration in Fig.2(a),we show the reflection spectra of dielectric compound gratings,whereγis-0.15 and 0.15,in Fig.2(c).Figure 2(c)shows that their reflection resonance wavelengths are not identical.Althoughγ=-0.15 andγ=0.15 have identical absolute values,different refractive indicesncan lead to different refractive indices of the dielectric compound grating,which affects the resonance wavelength of the grating.

Fig.2.Effect of asymmetric parameter γ on the grating reflection spectra.(a)Two-dimensional reflection spectra; (b)reflection spectra with γ being-0.05,-0.15 and-0.25;(c)reflection spectra with γ being-0.15 and 0.15.

Fig.3.Electric field distribution of the dielectric compound grating at the resonant wavelength with different values of material asymmetry parameter γ.(a)γ =-0.05;(b)γ =-0.15;(c)γ =-0.25.

There is no resonance in the dielectric compound grating atγ=0 because the grating is symmetric in this case,the radiation channels between the grating and the free space are closed, and no energy can leak from the dielectric compound grating into free space.Whenγ ̸=0,i.e.,the grating symmetry is broken and the radiation channels are formed,there is an energy leak from the grating into free space,and Fano resonances occur.[17,41,42]Asγgradually deviates from 0,which indicates that the asymmetry of the grating increases,more energy leaks into free space through the radiation channels.[17,41]Figure 3 shows the electric field distribution of the dielectric compound grating at the resonant wavelength,whereγis-0.05(n=3.4),-0.15 (n=3.3) and-0.25 (n=3.2).Whenγ=-0.05, the electric field energy of the dielectric compound grating is significantly higher than that ofγ=-0.15 andγ=-0.25.This is because a smallerγcorresponds to a lower degree of asymmetry of the grating, and less energy leaks out through the radiation channels.

In Fig.2, as the asymmetric parameterγgradually deviates from 0, the Fano resonance linewidth widens, which decreases theQfactor.Because theQfactor significantly affects the sensor sensitivity and GH shift,[13,14,43]the effect ofγon theQfactor of the dielectric grating QBIC resonance must be studied.Because the QBIC resonance is a Fano-like resonance, we fit the resonance based on the classical Fano formulation.The classical Fano[44]formula is given by

whereω0is the resonant frequency,Γis the resonant linewidth,T0is the background scattering parameter,A0is the coupling coefficient between continuous and discrete states andqis the Breit-Wigner-Fano parameter,which determines the asymmetry of the resonant spectral lines.According to Eq.(4), Fano resonances with different material asymmetric parameters can be fitted to obtainQ=ω0/Γ.Figure 3(a)shows the resonance fitting curve of the dielectric compound grating whenγ=-0.15,which is consistent with the simulation results,whereω0=1.414×1015s-1,Γ=3.5×1011s-1andQ=ω0/Γ=4.04×103.

Based on the above analysis, theQfactor for different material asymmetry parametersγcan be obtained by fitting simulated spectra using Eq.(4), and Fig.3(b) shows the specific results.In Fig.3(b),theQfactor of the QBIC resonance rapidly decreases whenγgradually deviates from 0.To obtain a largerQfactor,a smallerγis favored,but to ensure that the maximum GH shift is located at the reflection peak with perfect reflection, we must consider Fig.2(a) when determiningγ.In Fig.2(a),whenγis too small,it is difficult for the reflection resonance reflectance to reach 1.Therefore,the minimum value ofγis determined as-0.05(n=3.4).In this case,the QBIC resonance improves the GH shift and ensures that the maximum GH shift is located exactly at the reflection peak with unity reflectance.The material, whose refractive index is 3.4 near the resonance wavelength, is GaAs(see Fig.6 for details).

Fig.4.Fano fitting and the effect of material asymmetry parameter γ on the quality(Q)factor(w=200 nm,P=1200 nm).(a)Results of Fano fitting for γ =-0.15;(b)effect of γ on Q.

To achieve differentQfactors, geometric asymmetric gratings require precise adjustment of the geometric asymmetric parameters, but there may be errors in the fabrication process, resulting in poor controllability.However, the proposed material asymmetric grating method must only prepare the grating with the required refractive index to achieve the desiredQfactor without precisely adjusting the geometrical parameters, which has strong controllability.In the future,the grating with refractive indexncould also be replaced with phase-change material[45]or Dirac semimetal,[46]and the dynamic tuning of theQfactor can be realized through current injection or the photothermal effect.This would mean that the requiredQfactor could be obtained without changing the grating material through semiconductor fabrication technology.This would also be conducive to achieving the dynamic tunability of the GH shift without changing the dielectric compound grating.

3.2.Effect of the grating parameters on the QBIC resonance

In addition to material asymmetry parameterγ, the grating structural parameters affect the QBIC resonance.We calculated the effects of grating widthwand grating periodPon the dielectric grating reflection spectra, as shown in Fig.5.In Fig.5, the QBIC resonance wavelengths are redshifted whenworPincreases.The phenomenon can be explained by combining the grating guided-mode resonance and the equivalent medium theory.According to the literature,[15,30,39]the symmetry-protected BIC of the dielectric compound grating proposed in this paper originates from the guided-mode resonance.Because there is dielectric grating, the incident light and waveguide layer can be effectively coupled to cause guided-mode resonance.If the incident light and waveguide layer can be effectively coupled, the wave vector matching condition should satisfy the following conditions:[45]

whereneffis the equivalent refractive index of the dielectric compound grating,k= 2π/λris the incident wave vector,whereλris the guided-mode resonance wavelength;jis the integer order,andkx=ksinθis the component of the incident wave vector in thexdirection.From Eq.(5),the guided mode resonance wavelength can be deduced as[47]

According to Eq.(3), the dielectric compound grating refractive index is the function ofwandP.For the convenience of analysis, the influence ofwandPon the equivalent refractive indexneffof dielectric gratings will be analyzed using the equivalent medium theory.[39]From the equivalent medium theory,[48]the increase in grating widthwwill increase the grating equivalent refractive indexneff.According to Eq.(6),the increase inwwill redshift the guided-mode resonance wavelength.From the equivalent medium theory,Pweakly affects the equivalent refractive indexneffof the dielectric compound grating.However, Eq.(6) shows thatPis directly related to the guided-mode resonance wavelengthλr,so an increase inPwill inevitably redshiftλr.

Fig.5.Effect of the grating parameters on the QBIC resonance (θ =6°).(a)Grating width(P=1200 nm);(b)grating period(w=200 nm).

3.3.GH shift enhancement

In this section, we use the QBIC resonance excited by material asymmetry to achieve GH shift enhancement (w=200 nm,P=1200 nm).The material asymmetric parameterγis set at-0.05 (n=3.4),-0.15 (n=3.3) and-0.25(n= 3.2).Figure 2(b) shows that the corresponding QBIC resonance wavelengthλ(θ=6°) ofγ=-0.05,-0.15 and-0.25 are 1335.14 nm,1332.39 nm and 1330.26 nm,respectively,which correspond to the following three existing materials GaAs,AlSb,InP,respectively.Their refractive indices at resonant wavelengthsλare approximately[49-51]3.4, 3.3 and 3.2, as shown in Fig.6.These three materials are common semiconductor materials with mature fabrication and etching technology.

We calculated their reflection angle spectra by fixing the incident wavelength (λ), as shown in Fig.7.By comparing Fig.2(b) with Fig.7, we find that a narrower reflectance spectral linewidth corresponds to a narrower reflectance angle spectral linewidth.In Fig.7, the resonant wavelength reflectance reaches 1 nearθ=6°.According to the stationary phase method,when the incident light has a sufficiently wide beam waist,the GH shift can be expressed as[31]

whereφis the reflection phase.From Eq.(7),the GH shift is proportional to the partial derivative of the reflection phase to the incident angle.Therefore,to obtain the GH shift for differentγvalues, we calculated the reflection phase angle spectra based on the resonant wavelength, as shown in Fig.8.Whenγdecreases from-0.05 to-0.25, the slope of the reflection phase nearθ=6°gradually decreases.

Based on Eq.(3) and Fig.8, the GH shifts of differentγvalues are calculated and shown in Fig.9.The GH shift can reach-980λ0whenγ=-0.05, but the GH shift is only-180λ0whenγ=-0.25.Therefore,when the dielectric compound grating is composed of Si and GaAs materials,the GH shift is maximized.Whenγdecreases from-0.05 to-0.25,the GH shift rapidly decreases.This is because whenγis-0.05 and-0.25,theQfactor that corresponds to the dielectric compound grating is approximately 2.8×104and 1×103,respectively.TheQfactor affects the change in the reflection phase of the dielectric grating near the resonance angle.A largerQfactor corresponds to a more drastic change in the reflection phase.[16-18]Therefore, the GH shift is extremely high whenγ=-0.05,and whenγ=-0.25,the GH shift will sharply decrease.

Fig.6.Refractive indices of GaAs(a);AlSb(b);InP(c)at different wavelengths.

Fig.7.Reflection angle spectrum for different material asymmetry parameters γ.

Fig.8.Reflection phase angle spectrum for different material asymmetry parameters γ.

Fig.9.Goos-H¨anchen(GH)shift angle spectrum for different material asymmetry parameters γ.

The guided-mode resonance under oblique incidence is divided into two resonances of±1 order.This paper studies the +1 order resonance, and the propagation constant of the guided mode is negative, which will cause backward propagation resonant guided mode and result in a negative GH shift.[13,14]Currently, there are several studies on improving the positive GH shift based on QBIC,[13,36,37]but there are few studies on enhancing the negative GH shift.Therefore,in this paper, we choose to study the +1 order guided-mode resonance to enhance the negative GH shift.

To explore the effect of the grating structural parameters on the GH shift, we calculated the GH shift for different grating periodsPand grating widthsw, as shown in Fig.10(γ=-0.05).The GH shift can clearly be tuned by changingPorw,but the effect is not as significant as that ofγbecause the main factor that affects theQfactor is the material asymmetry parameterγ.WhenPandwchange, the resonance angle that corresponds to the maximum GH shift shifts.WhenPincreases from 1195 nm to 1205 nm, the resonance angle that corresponds to the maximum GH shift shifts from 6.02°to 6°.Whenwdecreases from 205 nm to 195 nm,the resonance angle that corresponds to the maximum GH shift shifts from 6.02°to 6°.

The dielectric compound grating has a highQfactor Fano resonance atγ=-0.05 (w= 200 nm,P= 1200 nm), so it can be used as a high-sensitivity refractive index sensor in climate monitoring and biochemical detection.Most refractive index sensors are based on the change in resonance wavelength,[52,53]whereas our sensor is based on the change in GH shift,which will greatly improve the sensor sensitivity.As shown in Fig.11,whennairincreases from 1 to 1.0001,the GH shift changes by 190µm atθ=6.01°.The refractive index sensitivity isS=ΔDGH/Δnair=1.9×106µm/RIU,which is significantly higher than those in Refs.[50-54].However,the sensitivity in Ref.[37] is one order of magnitude higher than ours because the presence of a substrate in our structure introduces an asymmetry in thez-direction,and the symmetryprotected BIC is extremely sensitive to symmetry-breaking perturbations.[55]In reality, to fabricate substrate-free metasurface is truly a challenge.

Fig.10.Effect of grating period P and width w on the Goos-H¨anchen shift: (a)P(w=200 nm);(b)w(P=1200 nm).

Fig.11.Effect of the environmental refractive index on the Goos-H¨anchen shift of dielectric compound gratings.

4.Conclusion

In this paper, the QBIC resonance with a highQfactor and high reflection was realized by breaking the material symmetry of the dielectric compound grating, and the QBIC resonance was applied to improve the GH shift.The effects of material asymmetry parameterγand grating structural parameters on theQfactor and resonance wavelength were analyzed based on the finite-difference time-domain method.γis the key factor that determines theQfactor.Whenγdecreases,theQfactor will sharply drop.The QBIC resonance wavelength is modulated by the grating width and period, and it will be redshifted when the grating width and period increase.The reflection angle spectra and reflection phase angle spectra of dielectric compound gratings were calculated forγ=-0.05(n=3.4),-0.15 (n=3.3) and-0.25 (n=3.2).A smaller material asymmetry parameter corresponds to a narrower reflection angle spectrum and a more drastic change in the reflection phase angle near the resonance angle.The material asymmetry-based QBIC was applied to enhance the GH shift.Based on the stationary phase method,the GH shift can reach-980 times the resonance wavelength whenγ=-0.05 (nis GaAs material),and the maximum GH shift corresponds to reflectance 1.Finally,using the dielectric grating as a refractive index sensor,the sensitivity of the refractive index sensor can reach 1.9×106µm/RIU.

Acknowledgements

Project supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No.LQ23F040001),the National Natural Science Foundation of China (Grant No.12204446), the Public Welfare Technology Research Project of Zhejiang Province (Grant No.LGC22E050006),and the Quzhou Science and Technology Project of China(Grant No.2022K104).