⇦Chinese Physics Letters, 2019, Vol. 36, No. 6, Article code 064202 Tunable Bistability in the Goos–Hänchen Effect with Nonlinear Graphene * Binbin Liu (刘彬彬)1,2, Pujuan Ma (马普娟)1,2, Wenjing Yu (於文静)1,2, Yadong Xu (徐亚东)1,2**, Lei Gao (高雷)1,2** Affiliations 1School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006 2Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006 Received 3 March 2019, online 18 May 2019 *Supported by the National Natural Science Foundation of China under Grant No 11774252, the Natural Science Foundation of Jiangsu Province under Grant No BK20161210, the Qing Lan project, the '333' project under Grant No BRA2015353, and the PAPD of Jiangsu Higher Education Institutions.
**Corresponding author. Email: leigao@suda.edu.cn; ydxu@suda.edu.cn Citation Text: Liu B B, Ma P J, Yu W J, Xu Y D and Gao L et al 2019 Chin. Phys. Lett. 36 064202    Abstract We present a planar model system of a silica covered with a monolayer of nonlinear graphene to achieve a tunable Goos–Hänchen (GH) shift in the terahertz range. It is theoretically found that the transition between a negative shift and a large positive one can be realized by altering the intensity of incident light. Moreover, by controlling the chemical potential of graphene and the incident angle of light, we can further control the tunable GH shift dynamically. Numerical simulations for GH shifts based on Gaussian waves are in good agreement with our theoretical calculations. DOI:10.1088/0256-307X/36/6/064202 PACS:42.65.Pc, 73.20.Mf © 2019 Chinese Physics Society Article Text The Goos–Hänchen (GH) effect refers to a linearly polarized light undergoing a spatially lateral displacement from the position predicted by geometry optics, when reflected at the interface of two media. It was observed experimentally for the first time by Goos and Hänchen in 1947.[1,2] Since then, it has been explored widely in theory and experiment,[3,4] in various structures and media including magneto-optical materials[5] and optical systems with parity-time symmetry.[6] With the advent of optical lasers, the GH effect has a significant role in applications such as microscopes[7] and temperature sensors.[8] Actually there have been many studies on the GH effect, featuring two points. In most reported results, the GH shift is settled in fixed geometry, and thereby cannot be effectively tuned through external fields, such as electric fields. Most studies focus their working wavelength on visible/infrared light and microwave frequencies. Few studies involve terahertz (THz) waves. Explorations of THz waves and associated applications are scarce, owing to a lack of THz sources/detections. In recent years, THz waves have attracted attention due to their unique characteristics including low photon energy, which exhibit great advantages over x-rays in the applications of imaging, spectroscopy and sensors.[9] Given these facts, it is of great interest to explore and demonstrate tunable GH shifts in THz regions. For instance, using multilayer structures, the GH shift in reflected light can be tuned by controlling the external bias voltage. Alternatively, utilizing the metal aluminum gives rise to a temperature-dependent GH shift in the reflected light from aluminum.[10] In this Letter, we propose a simple route to realize a tunable GH shift in the THz range. The geometry is a silica covered with a nonlinear monolayer graphene (see Fig. 1), based on two motivations. On the one hand, the designed structure should be kept as simple as possible, for easy integration with other compacted components. Then two-dimensional (2D) materials are considered to simplify the geometry. On the other hand, the GH shift should be controlled by light. The optical bistability (OB) effect provides a way of controlling light with light. Fortunately, graphene has a strong nonlinear response[11] in the terahertz regime, because of its large electron mobility.[12] Studies of nonlinear graphene have produced interesting phenomena, such as nonlinear surface plasmons[13] and the OB effect.[14] Here we show that the OB featured in nonlinear graphene can lead to a tunable GH shift with light.
cpl-36-6-064202-fig1.png
Fig. 1. Schematic diagram of the GH effect in the light reflected by a silica covered with a graphene sheet. The dotted line with arrow denotes the reflected beam predicted by the geometry optics, and the solid line stands for the practical reflected beam.
Figure 1 shows the studied geometric configuration, a silica of relative permittivity $\varepsilon_{2} =2.25$ covered with a single-layer graphene sheet located at a position $z=0$. The optical properties of graphene are described by the Kerr nonlinear surface conductivity $\sigma_{\rm g} =\sigma_{0} +\sigma_{3} |{\boldsymbol{E}}|^{2}$,[15,16] with the linear term $\sigma_{0} =\sigma_{\rm intra} +\sigma_{\rm inter}$ ($\sigma_{\rm intra}$ and $\sigma_{\rm inter}$ describe the intraband and interband contributions, respectively). From the Kubo formalism,[17] we have $$\begin{align} \sigma_{\rm intra} =\,&\frac{ie^{2}k_{\rm B} T}{\pi \hslash^{2}(\omega +i\gamma)}\Big[\frac{\mu_{\rm c}}{k_{\rm B} T}\\ &+2\ln \Big(\exp \Big[-\frac{\mu_{\rm c}}{k_{\rm B} T}\Big]+1\Big)\Big],~~ \tag {1a}\\ \sigma_{\rm inter} =\,&\frac{ie^{2}}{4\pi \hslash}\ln \Big[\frac{2\mu_{\rm c} -(\omega +i\gamma)\hslash}{2\mu_{\rm c} +(\omega +i\gamma)\hslash}\Big],~~ \tag {1b} \end{align} $$ where $e$ is the charge of the electron, $\hslash$ is the reduced Planck constant, $\mu_{\rm c} =\hslash v_{\rm f} \sqrt {n_{\rm c} \pi}$ is the chemical potential of graphene, $\gamma$ is the scattering rate, $v_{\rm F}$ is the Fermi velocity of electrons, $k_{\rm B}$ is Boltzmann's constant, and $T$ is temperature. The chemical potential $\mu_{\rm c}$ can be tuned by changing the carrier density $n_{\rm c}$.[18] When $\hslash \omega /\mu_{\rm c} < 2$, the interband transitions are forbidden, which produces a negligible scattering rate (i.e., $\gamma \approx 0)$ and negligible $\sigma_{\rm inter}$ compared with the intraband term. In addition, at room temperature ($T=300$ K), $\mu_{\rm c} \gg k_{\rm B} T$. Under these conditions, we can obtain the simplified forms of linear conductivity $\sigma_{0}$ and Kerr nonlinear surface conductivity $\sigma_{3}$ of graphene as follows:[15] $$\begin{align} \sigma_{0} =\,&ie^{2}\mu_{\rm c} /\pi \hslash^{2}\omega,\\ \sigma_{3} =\,&-i9e^{4}v_{\rm F}^{2}/8\pi \mu_{\rm c} \hslash^{2}\omega^{3}.~~ \tag {2} \end{align} $$ Consider a transverse magnetic (TM) polarized wave (with its magnetic field only along the $y$ direction) obliquely incident from air to the medium 2 (silica with graphene) with the incident angle $\theta$. After reflection, the electromagnetic (EM) fields in air are expressed as $$\begin{align} H_{1y} =\,&\frac{\omega \varepsilon_{0}}{k_{1z}}E_{0} [{\exp (ik_{1z} z)+r\exp (-ik_{1z} z)} ]\exp (ik_{x} x),~~ \tag {3a}\\ E_{1x} =\,&E_{0} [ {\exp (ik_{1z} z)-r\exp (-ik_{1z} z)} ]\exp (ik_{x} x),~~ \tag {3b} \end{align} $$ where $r$ is the reflection coefficient, $({k_{x}, k_{1z}})=({k_{0}\sin \theta, k_{0} \cos \theta})$, and $E_{0}$ is the amplitude of the incident electric field. Inside the medium 2, the EM fields are $$\begin{align} H_{1y} =\,&\frac{\omega \varepsilon_{0} \varepsilon_{2}}{k_{2z}}tE_{0} \exp (ik_{2z} z)\exp (ik_{x} x),~~ \tag {4a}\\ E_{2x} =\,&tE_{0} \exp (ik_{2z} z)\exp (ik_{x} x),~~ \tag {4b} \end{align} $$ where $k_{2z} =\sqrt {\varepsilon_{2} k_{0}^{2} -k_{x}^{2}}$. The continuous boundary conditions due to graphene are $$\begin{align} E_{1x} =E_{2x}, ~H_{1y} -H_{2y} =\sigma_{\rm g} E_{2x}.~~ \tag {5} \end{align} $$ Then, the reflection coefficient can be obtained as $$\begin{align} r=\frac{\varepsilon_{2} k_{1z} -\varepsilon_{1} k_{2z} +\sigma_{\rm g} k_{2z} k_{1z} /\omega \varepsilon_{0}}{\varepsilon_{2} k_{1z} +\varepsilon_{1} k_{2z} +\sigma_{\rm g} k_{2z} k_{1z} /\omega \varepsilon_{0}}.~~ \tag {6} \end{align} $$ It is clearly shown that comparing to the conventional case without graphene, the introduction of graphene leads to an extra term $\sigma_{\rm g} k_{2z} k_{1z} /\omega \varepsilon_{0}$ in the formula, which will greatly modify the reflection properties. Furthermore, the GH shift in the reflected wave is given by[19] $$\begin{align} D=-\frac{\lambda}{2\pi \cos \theta}\frac{d{\rm Im}[\ln (r)]}{d\theta},~~ \tag {7} \end{align} $$ where $\lambda$ is the wavelength of the incident wave in air, and ${\rm Im}[\ln (r)]$ denotes the phase of the reflection coefficient. Substituting Eq. (6) into Eq. (3) produces the amplitude of the local electric field inside graphene, $$\begin{align} E_{x| {z=0}} =\,&E_{0}(2\varepsilon_{1} k_{2z})/[\varepsilon_{2} k_{1z} +\varepsilon_{1} k_{2z} +(\sigma_{0}\\ &+\sigma_{3} |E_{x| {z=0}}|^{2})k_{2z} k_{1z} /\omega \varepsilon_{0}].~~ \tag {8} \end{align} $$
cpl-36-6-064202-fig2.png
Fig. 2. The reflection $|r|$ (a) and the phase $\phi$ (b) versus the incident angles for different chemical potentials. (c) The corresponding results of the GH shift $D$. (d) The numerical simulated results for GH effect at $\theta =45^{\circ}$ and $\mu_{\rm c} =0.35$ eV in (c). The inset shows the magnetic field pattern for the incident Gaussian beam.
Let us firstly examine the linear case, i.e., $\sigma_{3} =0$. Figures 2(a) and 2(b) show $|r|$ and $\phi$ versus the incident angle for different $\mu_{\rm c}$. The working frequency is $f=\omega /2\pi =0.34$ THz.[20] It is well known that for a dielectric silica without graphene, there is a Brewster angle for the TM wave given by $\tan (\theta_{\rm B})=\sqrt {\varepsilon_{2}}$, at which the reflected light vanishes (see the red dashed line in Fig. 2(a)). The added graphene significantly modifies the angular reflectivity, and there is the pseudo-Brewster angle $\theta_{\rm pB}$, which corresponds to the minimal reflectance, as shown in Fig. 2(a). Moreover, as $\mu_{\rm c}$ grows, $\theta_{\rm pB}$ is shifted to a large angle. In addition, the minimal value of reflectance deviates from zero, and becomes significantly larger. In general, a giant GH shift can be observed at the pseudo-Brewster angle, but it is not true in this study. This is due to the fact that no steep variations in the phase emerge around the pseudo-Brewster angle for any of the chemical potential levels. Figure 2(c) shows the calculated GH shifts. It is found that the introduction of the chemical potential in graphene is helpful to enhance the GH shift. Moreover, GH shifts are negative, and are monotonously increased with the incident angle. This behavior is quite similar to that of metals[21] at optical frequencies. This similarity is due to the fact that graphene has large surface conductivity, featuring metal-like characteristics. These results and the considered configuration suggest a way to achieve a metal-like GH effect in THz regions. To confirm the above analysis, full wave simulations were carried out using COMSOL Multiphysics. In simulations, the incident plane wave is replaced by a Gaussian beam with a wide waist $2\lambda$, and $\theta =45^{\circ}$. The graphene layer is modeled as a 2D surface current sheet with complex surface conductivity, and its thickness is set to be 0.33 nm. The inset in Fig. 2(d) shows the simulated filed pattern of the magnetic field, from which the GH shift can be seen directly by comparing the beam centers of the reflection (the right dashed arrow) with that of the incidence (the left solid arrow). The GH shift is indeed negative. To quantify its amplitude, the inset in Fig. 2(d) shows the field distribution at a position close to the interface, i.e., $z=0.01\lambda$ (indicated by the dashed line in Fig. 2(d)). It is shown that the GH shift in simulations is $D\approx 0.05\lambda$, which agrees well with the theoretical value $D\approx -0.06\lambda$.
cpl-36-6-064202-fig3.png
Fig. 3. (a) The OB for the local field, (b) the threshold field $E_{0}$, (c) amplitude $|r|$ and (d) GH shift $D$ as a function of $E_{0}$, for different chemical potentials $\mu_{\rm c}$. Here $\theta =45^{\circ}$.
Now we explore tunable GH shifts controlled by light using nonlinear graphene. Firstly, we discuss the OB effect in nonlinear graphene. Figure 3(a) analytically displays the relationship between the local electric field inside graphene (i.e., $E_{x \vert z= 0})$ and the incident electric field (i.e., $E_{0})$, for different chemical potentials $\mu_{\rm c}$. In calculations, we choose $f=0.34$ THz and $\theta =45^{\circ}$. An optical hysteresis loop emerges for large $\mu_{\rm c}$. For instance, when $\mu_{\rm c} =0.35$ eV, as shown by the black line in Fig. 3(a), two pink arrows indicate such hysteresis loop with two threshold electric fields: one is the switching-up field of $E_{\rm 0,up} \approx 1.19$ MV/m, and the other is the switching-down field of $E_{\rm 0,d} \approx 1.05$ MV/m. This optical hysteresis loop will lead to the following process for different incident intensities. When the incident electric field increases from zero, the local field in graphene firstly increases continuously. When the incident electric field reaches the switching-up value (i.e., $E_{\rm 0,up} \approx 1.19$ MV/m) and surpasses it, $E_{x \vert z= 0}$ will suddenly change from a low value of 0.45 MV/m to a high value of 0.73 MV/m, as shown by the right arrow. Subsequently it increases monotonically with the incident electric field. In turn, if the incident field is decreased back from a large value (e.g., 2.0 MV/m) to zero, the local field firstly decreases, and then jumps down to the lower stable branch when the incident field reaches the switching-down value of $E_{\rm 0,d} \approx 1.05$ MV/m, as shown by the left arrow. The area enclosed by this loop is the bistable region, and its size is $\Delta E_{0} =E_{\rm 0,up} -E_{\rm 0,down} =0.14$ MV/m. In fact, the OB region is $\mu_{\rm c}$-dependent. Figure 3(b) displays both the switching-up and switching-down threshold values. There is a critical chemical potential $\mu_{\rm c} \approx 0.29$ eV, beyond which the pronounced OB appears, and the OB region broadens with $\mu_{\rm c}$. Meanwhile, both the switching-up and switching-down thresholds become larger for large $\mu_{\rm c}$. Actually, a large $\mu_{\rm c}$ will produce a declined $\sigma_{3}$, thereby weakening the nonlinear effect of graphene. This means that a greater field intensity is required to keep the hysteresis loop. The OB effect will produce similar traits in the reflection spectra, as displayed in Fig. 3(c). For instance, for $\mu_{\rm c} =0.35$ eV, when the incident electric field increases from zero and reaches $E_{\rm 0,up} \approx 1.19$ MV/m, the reflectivity suddenly drops from a high value of about 0.82 to 0.30 (see the right arrow). In turn, when the incident electric field decreases from a high value and arrives at $E_{\rm 0,d} \approx 1.05$ MV/m, the reflectivity transits from about 0.20 to 0.90 (see the left arrow). Figure 3(d) shows the calculated GH shifts. For low $\mu_{\rm c}$ ($\mu_{\rm c}=0.15$ eV), the GH shift is a single-value function and continuously changes with the incident intensity. However, for large $\mu_{\rm c}$, the OB will produce a part of the three-valued region, in which the transition may occur. For the case $\mu_{\rm c} =0.35$ eV, at low incident fields, the GH shifts are negative and small. As the incident field increases up to the upper threshold field, such negative shifts are enhanced. Then the continuously increasing electric field leads to a transition of the GH shift from a negative value to a large positive value (see the right arrow). In turn, if one decreases the incident field from a high value to zero, the shift increases firstly and then decreases to a large negative value. After the switching-down threshold field, a similar transition, as shown by the left arrow, happens for the GH shift. Note that the sign change of the GH shift, i.e., from negative to positive, has potential applications in designing devices.[22] The OB effect provides a mechanism for this aim. One of the advantages of using graphene is that the required threshold can be tuned by controlling the chemical potential. The OB behavior is also largely dependent on the incident angle $\theta$. As a consequence the tunable GH shift can be further realized by adjusting the incident angle. Here we also take $\mu_{\rm c} =0.35$ eV as an example for discussions. Figure 4(a) shows $|{E_{x \vert z=0}}|$ inside graphene as a function of $E_{0}$. It is shown that the increasing incident angle will lead to the shrunken OB region, and the OB fully disappears for large $\theta$. The inset in Fig. 4(a) shows the threshold values for different angles. It is evident that there is a critical incident angle $\theta_{\rm c} =63^{\circ}$. Note that $\theta$ has a big impact on the magnitude of the reflection and the GH effect. Therefore, a small incidence is preferred to obtain a large bistable region. Figure 4(b) shows the corresponding results of the reflection $|r|$, respectively. When $\theta < \theta_{\rm c}$, the minimal reflection decreases as $\theta$ increases. From Fig. 4(c), the OB for the GH shift exists for small $\theta$, but the magnitude of the GH shift is small. On the other hand, for large $\theta$, although the OB disappears, one can observe a large variation of the GH shift with $E_{0}$. Therefore, to achieve the required OB effect and the expected control of the GH shift, there is a trade-off between $\mu_{\rm c}$ or $\theta$ and the electric field, requiring an optimization process.
cpl-36-6-064202-fig4.png
Fig. 4. The influences of the incident angle on the OB effect (a), the reflection (b) and GH shifts (c). In (a)–(c), $f=0.34$ THz and $\mu_{\rm c} =0.35$ eV. (d) The comparisons between the results based on the plane wave (the black line) and the Gaussian-shaped incident beam (the dashed line) at a fixed incident angle $\theta =45^{\circ}$.
It should be noted that although the above discussions are based on the stationary-phase method,[19,23] all obtained results are valid for a Gaussian-shaped incident beam. Consider a Gaussian wave that is obliquely incident onto the studied configuration of Fig. 1, given by[24] $$\begin{alignat}{1} E_{r}|{_{z=0}}=\,&\frac{1}{\sqrt {2\pi}}\int\limits_{-\infty}^{+\infty} {r(k_{x})A(k_{x})\exp (ik_{x} x)dk_{x}},~~ \tag {9a}\\ A(k_{x})=\,&w_{x} \sec \theta \exp [-(w_{x}^{2} /2)(k_{x} -k_{x0})^{2}],~~ \tag {9b} \end{alignat} $$ where $r(k_{x})$ is the reflection coefficient, $k_{x}$ is the $x$-component of the incident wave vector, and $A(k_{x})$ is the angle spectrum of the Gaussian wave, with $k_{x0} =k_{0} \sin \theta$, $w_{x} =w_{0} \sec \theta$, and the waist width $w_{0}$. Based on the method in Ref.  [24], Fig. 4(d) presents the comparison between the theoretical and numerical results. It is shown that the analytic predictions are in good agreement with the numerical results. In conclusion, we have demonstrated that using nonlinear graphene, the GH shift can be effectively controlled through the incident intensity, based on the OB effect contributed by the nonlinear optics. In particular, the transition from a negative shift to a positive value, featuring the signature of sign change, can be observed by controlling the intensity of incident electric fields. Although all analysis in this work focuses on the lossless case, similar results including the OB effect and tunable GH shift can still be seen in nonlinear graphene with a moderated loss. Our proposal suggests an efficient way of tuning the GH shift controlled by light. References Ein neuer und fundamentaler Versuch zur TotalreflexionExtension of the Hartree method to strongly interacting systemsDirect measurement of the optical Goos-Hänchen effect in lasersMeasurement of the Nonlinear Goos-Hänchen Effect for Gaussian Optical BeamsMagnetic control of Goos-Hänchen shifts in a yttrium-iron-garnet filmLarge and tunable lateral shifts in one-dimensional PT-symmetric layered structuresDetection of subwavelength Goos–Hänchen shifts from near-field intensities: a numerical simulationHigh-sensitivity temperature sensor using the ultrahigh order mode-enhanced Goos-Hänchen effectCutting-edge terahertz technologyTemperature-dependent Goos–Hänchen shift in the terahertz rangeCoherent Nonlinear Optical Response of GrapheneElectric Field Effect in Atomically Thin Carbon FilmsElectrically tunable nonlinear plasmonics in graphene nanoislandsTunable Optical Bistability and Tristability in Nonlinear Graphene-Wrapped NanospheresOptical bistability of graphene in the terahertz rangeNonlinear optical response of a two-dimensional atomic crystalDyadic Green’s functions and guided surface waves for a surface conductivity model of grapheneSynthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequenciesUeber drehende Schwingungen eines StabesTunable THz optical bistability in graphene-based heterostructuresLarge negative Goos–Hanchen shift at metal surfacesGiant bistable shifts for one-dimensional nonlinear photonic crystalsNegative Lateral Shift of a Light Beam Transmitted through a Dielectric Slab and Interaction of Boundary EffectsLateral shift of the transmitted light beam through a left-handed slab
[1] Goos F and Hänchen H 1947 Ann. Phys. 436 333
[2] Goos F and Hänchen H 1958 Ann. Phys. 5 251
[3] Bretenaker F, Floch A L and Dutriaux L 1992 Phys. Rev. Lett. 68 931
[4] Emile O, Galstyan T, Le Floch A and Bretenaker F 1995 Phys. Rev. Lett. 75 1511
[5] Yu W J, Sun H and Gao L 2017 Sci. Rep. 7 45866
[6] Ma P J and Gao L 2017 Opt. Express 25 9676
[7] Madrazo A and Nieto-Veperinas M 1995 Opt. Lett. 20 2445
[8] Wang X, Yin C, Sun J, Li H, Wang Y, Ran M and Cao Z 2013 Opt. Express 21 13380
[9] Tonouchi M 2007 Nat. Photon. 1 97
[10] Zang M D, He T, Zhang B, Zhong L and Shen J 2016 Opt. Commun. 370 81
[11] Hendry E, Hale P J, Moger J, Savchenko A K and Mikhailov S A 2010 Phys. Rev. Lett. 105 097401
[12] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666
[13] Cox J D and de Abajo F J G 2014 Nat. Commun. 5 5725
[14] Zhang K, Huang Y, Miroshnichenko A E and Gao L 2017 J. Phys. Chem. C 121 11804
[15] Peres N M R, Bludov Y V, Santos J E, Jauho A P and Vasilevskiy M I 2014 Phys. Rev. B 90 125425
[16] Merano M 2016 Opt. Lett. 41 187
[17] Hanson G W 2008 J. Appl. Phys. 103 064302
[18] Gan C H, Chu H S and Li E P 2012 Phys. Rev. B 85 125431
[19] Artmann K 1799 Ann. Phys. 2 87
[20] Yu B B, Deng X H, Yuan J and Liao Q H 2015 J. Mod. Opt. 62 1650
[21] Leung P T, Chen C W and Chiang H P 2007 Opt. Commun. 276 206
[22] Hou P, Chen Y Y, Chen X, Shi J L and Wang Q 2007 Phys. Rev. A 75 045802
[23] Li C F 2003 Phys. Rev. Lett. 91 133903
[24] Chen X and Li C F 2004 Phys. Rev. E 69 066617