Zeng-Ping Su(蘇增平) Tong-Tong Wei(魏彤彤) and Yue-Ke Wang(王躍科)

1Optical Information Science and Technology Department,Jiangnan University,Wuxi 214122,China

2Optoelectronic Engineering and Technology Research Center,Jiangnan University,Wuxi 214122,China

Keywords: one-dimensional photonic crystals,graphene,topological interface states

1. Introduction

Photonic crystals (PhC), as an artificial periodic dielectric material with periodicity in one, two or all three orthogonal directions, have been intensively investigated since Yablonovitch[1]and John[2]independently proposed the concept of PhC in 1989.[3]Similar to the electronic band gaps in semiconductor, PhC can prohibit electromagnetic waves in some specific frequency zones called the photonic band gaps.[4–6]In recent years, the studies of photonic topological insulators have attracted much attention due to their extraordinary properties including the elimination of backscattering and immunity to structural disorder.[7–9]Particularly,topologically protected edge states in PhC show broad prospects in the application of integrated photonic circuits and ultrahighspeed information processing chips.[10]For one-dimensional(1D)PhC system,topological interface states(TIS)exist at the interface of the adjacent PhC with different topological properties. The existence of TIS can generate a sharp transmission peak within the photonic band gaps and a highly localized electric field around the interface,[11,12]which is promising to be used to enhance the light–matter interactions.[13]

Graphene has been demonstrated as a good candidate for photonic and optoelectronic devices due to its unique characteristics, including the ultrahigh charge carrier mobility, extremely wide spectral range,and dynamical tunability via external doping or bias voltage.[14]In the range from mid- to far-infrared, the optical properties of graphene are similar to those of Drude-type materials, and strong resonance absorption can be achieved through plasmonic resonance.[15–18]However, graphene is almost transparent with an absorptivity of 2.3% in the visible and near-infrared range,[19]which largely limits its substantial application in photon detection and photoelectric conversion.[20]To overcome such shortcomings, optical interface states, such as Tamm plasmon polaritons (TPP) and TIS, can be introduced to enhance the light absorption of monolayer graphene.[21–29]For TIS and TPP,strong field localization can be realized at interface, which plays an extremely important role in the enhanced absorption of graphene. The strong field localization strengthens the interaction between graphene and incident light; thus,the higher absorption efficiency can be achieved. Luet al. numerically and theoretically investigated high-efficiency light absorption of monolayer graphene via TPP in the near-infrared range.[21]Wanget al. achieved multichannel terahertz perfect absorber by TPP in the far-infrared range.[22]Linet al.theoretically presented a tunable light absorption of graphene using TIS in visible range.[28]Also in visible range, Huet al. theoretically accomplished dual-band absorption enhancement by strong coupling of TPP and TIS in 1D topological PhC heterostructure/Ag hybrid system.[24]Nevertheless, the near-infrared absorption enhancement with graphene through the coupled modes of TIS in all-dielectric topological PhC heterostructure system has not been reported.

In this work,we propose to achieve dual-channel absorption enhancement with graphene by the all-dielectric topological PhC heterostructure in the near-infrared range. The monolayer graphene is placed at the interface of adjacent PhC with opposite topological properties,where TIS can be excited.The dual-channel nearly perfect absorption can be obtained due to the coupled modes of TIS. We numerically and theoretically analyze the influence of pairs number of PhC in the center and incident angle(TE and TM polarizations)on the dual-channel absorption curves based on finite element method(FEM)and transfer matrix method (TMM), respectively. Besides, the dual-channel optical switches can be realized by controlling the Fermi level of graphene.

2. Results and discussion

As shown in Fig.1,the structure used to realize the coupled modes of TIS is composed of PhC-F(alternatingNF(=4)pairs of Si/SiO2/Si)/graphene/PhC-M (alternatingNM(= 2)pairs of SiO2/Si/SiO2)/graphene/PhC-B(alternatingNB(=8)pairs of Si/SiO2/Si)on a GaAs substrate. A plane wave of TM polarization(magnetic field is perpendicular to thex–zplane)or TE polarization (electric field is perpendicular to thex–zplane)is incident on the proposed structure from surrounding medium (Si) along the positivezaxis with an incident angleθinx–zplane. Here,the parameters of the proposed structure area1=690 nm,b1=822 nm,a2=205 nm,b2=170 nm,andds=2000 nm for GaAs substrate.The parameters of PhCF and PhC-B are the same,except for the number of pair(NFandNB). The refractive indices of Si, SiO2, and GaAs arena=2.82,nb=1.46 andns=3.4, respectively.[10,30]We set the thickness of monolayer graphenedg=0.35 nm. The surface conductivity of monolayer graphene isσg, which is calculated according to the Kubo formula[25,31]

whereEfis the Fermi level,τis electron–phonon relaxation time,eis the charge of an electron,kBis the Boltzmann’s constant,ωis the radian frequency,and ˉhis the reduced Planck’s constant. The permittivityεgof graphene can be written asεg=1+iσgη/(kdg),whereη(=377 ?)is the impedance of air. Here,the Fermi levelEfis 0.2 eV,Tis 300 K,relaxation timeτisμEf/(ev2f), DC mobilityμis 10000 cm2/V·s, and Fermi velocityvfis 1×106m/s.

Fig. 1. Schematic diagram of the 1D all-dielectric PhC heterostructure consisting of PhC-F/graphene/PhC-M/graphene/PhC-B on a GaAs substrate. In each unit cell,a single layer of SiO2 (or Si)is sandwiched between half layers of Si(or SiO2)for PhC-F(B)(or PhC-M).And the light illuminates from surrounding medium(Si)along the positive z axis.

Fig.2. Photonic band structure for PhC-F(B)(a)and PhC-M(b).

In this work,all numerical simulation results are obtained by the commercial software COMSOL Multiphysics based on FEM, and the periodic boundary condition are set in thexdirection. In addition, we theoretically analyze the propagation features of the proposed all-dielectric PhC heterostructure by TMM.[36]Figure 3(a) shows the absorption and reflection curves for TE polarization at normal incidence(θ=0?)with(solid lines)and without(dots)graphene layer.It can be found that two absorption peaks appear for the PhC heterostructure with graphene layer,corresponding to the absorption value of 96.7% at 1564.8 nm and 97.9% at 1632 nm. However, the absorption is almost zero, and the two reflection valleys also appear at 1564.8 nm and 1632 nm for the PhC heterostructure without graphene layer. It is believed that the two absorption peaks in the PhC heterostructure with graphene layer are attributed to the excitation of the coupled modes of TIS.And the full width at half maximum(FWHM)of two absorption peaks are 1.6 nm and 3 nm, respectively. According to the definition of quality factor (Q), i.e.,λ/?λ, we obtain the quality factors for two absorption peaks, which are 978 at 1564.8 nm and 544 at 1534.6 nm, respectively. To illustrate the coupled modes of TIS, the calculated electric field distribution ofEyand profile of|Ey|2at 1564.8 nm and 1632 nm are shown in Figs.3(b)and 3(c),respectively. Obviously,the field and energy are mainly localized near the interfaces of PhC-F/PhC-M and PhC-M/PhC-B, which are also the position of the graphene layers. In addition, the calculated electric field distribution ofEyis antisymmetric about the center of PhC-M (z=0 nm) in Fig. 3(b), and the calculated electric field distribution (Ey) is symmetric about the center of PhC-M (z=0 nm) in Fig. 3(c). Because the two TIS can be excited at the interfaces of PhC-F/PhC-M and PhC-M/PhCB, the TIS generated at the two interfaces can couple with each other, resulting in coupled modes of TIS, named antisymmetric mode(the absorption peak at 1564.8 nm)and symmetric mode (the absorption peak at 1632 nm), respectively.Due to the coupled modes of TIS,the highly localized electric field and energy at the interface enhance the interaction between light and graphene, and the dual-channel near-infrared absorption is achieved. Then, we study the influence ofNM(the pair numbers of PhC-M)on TIS-TIS coupling,as shown in Fig. 3(e). We calculate the absorption curves whenNMis 2,3, 4,7,and 10 based on FEM(black lines)and TMM(red dots). Clearly, asNMincreases, the absorption peaks corresponding to the coupled modes of TIS (anti-symmetric and symmetric modes) gradually move closer, and only a perfect absorption peak remains whenNM=10. Figure 3(d) shows electric field distribution ofEyand profile of|Ey|2at absorption peak (1601.7 nm) corresponding to the absorption value of 99.9%andQof 517 whenNM=10. For our proposed alldielectric PhC heterostructure with smallerNM,the two TIS at the two interfaces can more strongly interact with each other,which leads to greater splitting for the coupled modes of TIS in wavelength regime. On the contrary, the interaction between TIS at the two interfaces gradually weakens,the two coupled modes of TIS gradually move closer. WhenNM= 10, the interaction between TIS almost disappears and light is completely absorbed at the interface of PhC-F/PhC-M due to TIS,so only one absorption peak is retained. Thus, owing to the coupled modes of TIS, the dual-channel absorption enhancement with graphene can be achieved,and the dual-channel absorption enhancement can be adjusted by changingNM. WhenNMis large enough(NM=10),the single channel perfect absorption can also be achieved.

The effect of the Fermi level of graphene(Ef)on the dualband enhancement absorption is also investigated. The imaginary parts of permittivity of graphene with Fermi level and wavelength are calculated as shown in Fig. 4(a). Obviously,When Fermi level is around 0.4 eV, the imaginary parts of permittivity of graphene will rapidly change from 17 to 0 as Fermi level increases. It leads to a rapid decay of absorption.Fermi levelEfcan be actively changed by chemical doping and electrical or thermal stimulation.[37]Figures 4(b)and 4(c)depict absorption of numerical simulation as a function of the wavelength andEfat normal incidence(θ=0?)whenNM=2 and 4,respectively. It can be clearly observed that absorption does not have significant variations atEfof less than 0.4 eV,but whenEfis slightly greater than 0.4 eV,the absorption experiences a significant drop. This is consistent with the result obtained by Fig. 4(a). As shown in the insets in Figs. 4(b)and 4(c),whenEfincreases from 0.36 eV to 0.42 eV,the peak absorption for wavelength 1564.8 nm, 1632 nm, 1591.3 nm and 1611.3 nm drops from 93.1% to 6.5%, 96.5% to 7.4%,90.1% to 6.8%, and 92.8% to 7.1%, respectively. It means that the dual-channel absorption can be rapidly adjusted at the same time by changingEfaround 0.4 eV. This phenomenon is valuable for many practical applications, including optical switches, modulators, etc. Besides, the working wavelength can also be selected by adjustingNM.

Fig.3. (a)Comparison of absorption and reflection curves with and without graphene for the proposed structure. The electric field distribution of Ey and profile of|Ey|2 at 1564.8 nm(b)and 1632 nm(c)with graphene when NM=2. (d)The electric field distribution of Ey and profile of|Ey|2 at 1601.7 nm with graphene when NM=10. (e)Absorption curves at NM=2,3,4,7,and 10 based on FEM(black lines)and TMM(red dots).

Fig.4. (a)Imaginary parts of permittivity spectra of graphene with Fermi level. Absorption as a function of the wavelength and the Fermi level of graphene(Ef)when NM=2(b)and 4(c). The insets of(b)show absorption versus Ef at 1564.8 nm and 1632 nm,respectively. The insets of(c)show absorption versus Ef at 1591.3 nm and 1611.3 nm,respectively.

Finally,the influence of the incident angleθon the dualchannel absorption is investigated for TE and TM polarization. Figures 5(a)and 5(c)show the absorption as a function of the wavelength and the incident angleθf(wàn)or TE and TM polarization,respectively. The results calculated by TMM are also shown in Figs. 5(a) and 5(c) by cyan dots. The geometric parameters of the structure are the same as those used in Fig.3(a). As shown in Figs.5(b)and(d),for TE and TM polarization,the absorption curves at the incident angleθis 0?,5?,10?,and 15?are calculated by FEM(solid lines)and TMM(dots), respectively. Obviously, it can be seen that the results obtained by FEM and TMM are identical. Due to the dependence of the gaps on the polarization and angle of incident light,[38]the change of the incident angleθwill have an impact on the overlap range of gaps for the TE or TM polarization.Here,For TE and TM polarizations,when the incident angleθvaries from 0?to 15?,the overlap range of gaps changes from 1500–1655 nm to 1380–1550 nm and 1500–1655 nm to 1400–1525 nm, respectively. Also, the blue shift for the coupled modes of TIS happens. For TE polarization, the wavelengths corresponding to the dual-channel absorption peaks vary from 1564.8 nm and 1632 nm to 1468.5 nm and 1525.4 nm.And the dual-channel absorption peaks both exceed 90% atθof less than 8?. For TM polarization,the wavelengths corresponding to the dual-channel absorption peaks vary from 1564.8 nm and 1632 nm to 1443.5 nm and 1509.6 nm. And the dual-channel absorption peaks both exceed 90%atθof less than 12?. Thus,the dual-channel absorption enhancement induced by coupled modes of TIS is robust and has a certain tolerance to incident angle for TE and TM polarization.

Fig.5. Absorption as a function of the wavelength and the incident angle θ for TE(a)and TM(c)polarization, respectively. The cyan dots represent the results of TMM in(a)and(c). Absorption curves based on FEM(solid lines)and TMM(dots)at θ =0?,5?,10?,and 15?for TE(b)and TM(d)polarization.

3. Conclusion

In summary,an all-dielectric heterostructure consisting of PhC-F/graphene/PhC-M/graphene/PhC-B on GaAs substrate is proposed. TIS can be excited on the interfaces of PhCF/PhC-M and PhC-M/PhC-B,where the coupled modes of TIS are generated through coupling between TIS.Due to the existence of the coupled modes of TIS,the highly localized electric field and energy at the interface promote the strong interaction between light and graphene, the dual-channel nearinfrared absorption enhancement is realized. By increasingNM,the dual-channel nearly perfect absorption can be modulated to single channel perfect absorption. The dual-channel absorption drops sharply by fine-tuning the Fermi level of graphene around 0.4 eV.Besides,the dual-channel absorption can be modulated by changing angle of incident light(TE and TM polarization). The simple all-dielectric layered system of the proposed heterostructure makes it promising for applications in optical absorbers,switches,sensing and modulators.

Acknowledgement

This project was supported by Postgraduate Research& Practice Innovation Program of Jiangsu Province, China(Grant No.KYCX201929).