Switching between positive and negative group delay of the optical pulse refection from layer structures with a Graphene sheet

TL;DR

This study demonstrates how the reflection of optical pulses from layered structures with graphene can be tuned between positive and negative group delays by adjusting graphene's properties, enabling advanced control of pulse propagation.

Contribution

It introduces two mechanisms—resonances and surface plasmon excitations—for tunable control of optical pulse delays in graphene-layered systems, advancing optical device design.

Findings

Reflected group delays can be tuned from positive to negative.

Tuning Fermi energy and temperature controls delay properties.

Mechanisms involve resonances and surface plasmon excitations.

Abstract

In this paper, we investigate the propagation of the light pulse reflected from the layer system with a graphene layer. We show a tunable transition between positive and negative group delay of the optical pulse reflection in such a layered system controlled by the properties of the graphene layer, and reveal two mechanisms to control the propagation properties of the light reflected from such systems. It is demonstrated that the reflected group delays are greatly tunable from positive and negative values in both mechanisms of resonances and the excitations of the surface plasmon resonances, which are adjusted by tuning the Fermi energy and temperature of the graphene layer. Our results are helpful to control the pulse propagations and are useful for design of graphene-based optical devices.

Equations

$$r=\frac{r_{{}_{12}}+r_{{}_{\sigma }}{e}^{i2k_{2z}d}}{1+r_{{}_{12}}{r}_{{}_{\sigma }}{e}^{i2k_{2z}d}},$$

$${\tau }_r=\frac{d{\varphi }_r\left(\omega \right)}{d\omega }\text{,}$$

Full text

I Introduction

Controlling the group delay (or group velocity) of a light pulse has been extensively studied in both theories and experiments for many years Milonni2005 ; Chiao1997 ; Brillouin1960 ; Khurgin2009 . When group velocity is much smaller than the speed of light $c$ in vacuum, it often refers to slow light with a large value of group delay. The large positive group delay has potential applications in realizing optical delay lines Khurgin2009 and a high-efficiency memory for optical pulses Novikova2012 . The approaches for realizing this delay have been widely investigated in various circumstances, such as ensembles of warm atoms (i.e. Rb and Cs) in vapor cells Novikova2012 , photonic crystal waveguides Kurt2013 ; Monat2010 ; Schulz2010 , and multiple quantum wells Yan2013 . Meanwhile, when group velocity is possible to be much larger than $c$ or even becomes negative, it may be called as fast light with a very small or negative group delay. It should be emphasized that this group delay do not violate causality or special relativity because group velocity is not regarded as a signal velocity Chiao1997 . It has been observed in different systems including left-handed media Woodley2004 , the samples of GaP:N ChuandWong1982 , quantum wells Vetter2001 , double-Lorentzian fiber grating Longhi2002 , weakly absorbing slabs Wang2006 , atomic media Akulshin2010 , and ruby Gao2010 . Furthermore, in some system or media, such as fibers Arrieta-Yanez2010 ; Thevenaz2008 , semiconductor waveguides Mork2010 ; Mork2009 , nonlinear wave-mixing processes Bortolozzo2010 , solids at room temperature Zhang2008 ; Bigelow2004 , and gain slabs Wang2014 , the two types of group delay can been demonstrated simultaneously. Recently, with the emergence of new materials, controlling the group delay in other structures and media are receiving more and more attention.

Graphene, a one-atom-thick allotrope of carbon, is focused extensively in materials science and condensed-matter physics Bonaccorso2010 ; Geim2007 . It has linear dispersion relation of electronic states characterized by conical and valence bands joined together at Fermi level (the so called Dirac point), and the Fermi level can be controlled by application of external or magnetic fields Castro2009 ; Christensen2012 ; Gusynin20061 ; Gusynin20062 ; Falkovsky2007 ; Vakil2011 ; Zi2013 . Due to the tunability, graphene shows many potential applications in graphene-based nano-electronic and opto-electronic devices Novoselov2004 ; Castro2009 . Several research papers have been focused on the manipulation of the light propagation in the system with graphene by controlling the conductivity of graphene. In 2015, Hao et al. revealed that graphene exhibits much stronger slow light capability than other materials Hao2015 , and a large delay-bandwidth product has been obtained in graphene-based waveguide. Meanwhile, Lu et al. designed a kind of plasmonic structure consisting of a monolayer graphene to slow down and trap the light in the mid-infrared region Lu2015 . Shi et al. found that the plasmonic modes in the graphene nanostructure can be confined to a spacial size that is hundreds of times smaller than their corresponding wavelength in vacuum Shi2013 . Li et al. investigated the graphene ribbon waveguide and achieved an outstanding plasmonically induced transparency window with a group time up to 0.28ps Li2015 . On the other hand, in 2014 Jiang 2014 , Jiang et al. investigated the negative group delay of the TE-polarized beam reflected from a Fabry-Perot cavity with the insertion of the graphene. Next year, they found that a fast pulse reflection can take place from the graphene covered lossless dielectric slab Jiang 2015 . It is possible to realize the large positive and negative group delay in the same structure with the help of the graphene layer. In addition, the Otto configuration combined with graphene have been investigated, and the transverse magnetic surface plasmons Mendieta 2014 , perfect terahertz absorption LYJiang 2016 , and surface modes of transverse electric polarization Mason 2014 ; Mendieta 2015 ; Menabde 2016 have been proved and studied in succession in such a configuration.

Motivated by these studies, we have theoretically considered the light reflected from the layer configuration that incorporates graphene. Two mechanisms that realize the group delay of the reflected pulse to be control from positive to negative values, or vise verse, are demonstrated. The first is related to resonances occurred when the incident angle is smaller than the total internal reflection angle. The second one is related to the excitation of surface plasmon happened when the incident angle is larger than the total internal reflection angle. Both of cases lead to a distinct variation of the group delays of the reflected light pulse, and it can be controlled by adjusting the Fermi energy and the temperature of the graphene sheet. Moreover, the structural parameters such as the position of the graphene layer and angle of incidence can also provide an effective method to control the reflected light pulse. Our results may have potential application in graphene-based optical technologies and information processing.

II Model and Calculation

Let a light pulse with a carrier frequency $\omega$ inside the dielectric medium $\varepsilon_{1}$ be incident on the slab system containing a graphene layer. As shown in Fig. 1(a), the graphene layer is placed inside the dielectric medium $\varepsilon_{2}$ with a distance $d$ to the interface between dielectric media $\varepsilon_{1}$ and $\varepsilon_{2}$, and $\theta$ is the incident angle of light. Here it should be emphasized that since $\varepsilon_{1}>\varepsilon_{2}$, when the total internal reflection occurs, in this sense, the structure can be seen as Otto configuration similar to that in Ref. Bludov2013 . The surface conductivity $\sigma$ of graphene is usually given by Kubo formula. In the low temperature limit $k_{B}T\ll E_{F}$ and the zero collision rate ($\tau^{-1}=0$), the surface conductivity can be expressed as Falkovsky2007 $\sigma=\sigma_{\text{Intra}}+\sigma_{\text{Inter}}$, where $\sigma_{\text{Intra}}=\frac{i8\sigma_{0}k_{B}T}{\pi\hbar\omega}\ln\left(2\cosh\frac{E_{F}}{2k_{B}T}\right)$ is the intraband electron transition contribution, and $\sigma_{\text{Inter}}=\frac{\sigma_{0}}{2}+\frac{\sigma_{0}}{\pi}\arctan(\frac{\hbar\omega-2E_{F}}{2k_{B}T})-\frac{i\sigma_{0}}{2\pi}\ln[\frac{\left(\hbar\omega+2E_{F}\right)^{2}}{\left(\hbar\omega-2E_{F}\right)^{2}+\left(2k_{B}T\right)^{2}}]$ is the interband electron transition contribution. Here $\sigma_{0}=\frac{e^{2}}{4\hbar}$, $e$ is the charge of an electron, $\hbar$ is the reduced Plank’s constant, $k_{B}$ is the Boltzmann constant, $T$ denotes the temperature, and $E_{F}$ is the Fermi energy which may be electrically adjusted by the applied gate voltage. Clearly the real part of $\sigma$ is determined by $\sigma_{\text{Inter}}$ while its imaginary part is determined by both $\sigma_{\text{Intra}}$ and $\sigma_{\text{Inter}}$. From Figs. 1(b) and 1(c), it is seen that the optical conductivity of graphene with frequency is shifted when the Fermi energy increases. In addition, the optical conductivity of graphene is also affected by temperature. Meanwhile, it is also noted that the real part of $\sigma$ (related to the absorption) changes dramatically near the energy of light frequency close to $2E_{F}$. With these properties, it is expected that the propagation properties of an optical structure containing graphene may be controlled by both these external and structural parameters.

Generally speaking, there are two methods in treating graphene as an element of a layered structure for optical applications. On one hand, the graphene can be considered as a zero-thickness layer characterized by the two dimensional conductivity Zi2013 ; Jiang 2014 ; Jiang 2015 ; Mendieta 2014 ; LYJiang 2016 . On the other hand, it can be seen as a thin layer with effective dielectric constant Othman2013 ; EI-Naggar2015 . For our purposes, using one of these methods is only for theoretical convenience and the same results are expected by another method. Here we treat graphene as a zero-thickness layer. According to the Maxwell’s equation and the boundary conditions, the reflection coefficient $r$ for TM-polarization is given by

[TABLE]

where $r_{{}_{12}}=\frac{\varepsilon_{2}k_{1z}-\varepsilon_{1}k_{2z}}{\varepsilon_{2}k_{1z}+\varepsilon_{1}k_{2z}}$ is Fresnel reflection coefficient between $\varepsilon_{1}$ and $\varepsilon_{2}$, $r_{{}_{\sigma}}=\frac{\sigma k_{2z}}{2\varepsilon_{0}\varepsilon_{2}\omega+\sigma k_{2z}}$ is the Fresnel reflection coefficient for the interface of a graphene layer inside $\varepsilon_{2}$, $\varepsilon_{0}$ is the permittivity of vacuum, and $k_{jz}=\left(k_{0}^{2}\varepsilon_{j}-k_{y}^{2}\right)^{1/2}$ $(j=1,2)$ is the $z$ component of the wave vector inside the $j$th medium with $k_{y}=k_{0}\sqrt{\varepsilon_{1}}\sin\theta$ and $k_{0}=\frac{\omega}{c}$. For the narrow-spectrum incident pulse, i.e., $\Delta\omega\ll\omega$ ($\Delta\omega$ is the spectrum width), the group delay of the reflected pulse can be calculated by Sanchez2010

[TABLE]

where $\phi_{r}$ is the phase of the reflection coefficient. In our calculation, without loss of generality, we assume $\varepsilon_{1}=2.25$ and $\varepsilon_{2}=1$ for vacuum (or air). This means that the Brewster angle $\theta_{B}$ of the system is $\theta_{B}=33.69{{}^{\circ}}$, and the critical angle $\theta_{c}$ of total reflection is $\theta_{c}=41.8{{}^{\circ}}$.

III Numerical Results and Discussions

First we consider the case of normal incidence, and the results presented here can be extended to the angles of incidence $\theta<\theta_{c}$. In Fig. 2, we show the typical properties of light pulse reflected from the layered system containing a graphene layer for different values of $d$ ($d=30\mu$m, 40$\mu$m and 90$\mu$m) in the case of normal incidence. It is clear that the reflection dips in Fig. 2(a) are observed as a result of the structural resonance. For convenience, we use $\omega_{R,j}$ to denote the resonant frequencies with $j=1,2,3\cdots$. In Fig. 2(b), we compare the magnitudes of Fresnel reflection coefficients $\left|r_{12}\right|$ and $\left|r_{\sigma}\right|$. In our calculation, $|r_{12}|=0.2$ for normal incidence, whereas $|r_{\sigma}|$ decreases with the increase of frequency. Thus there is a critical frequency $\omega_{c}$ which makes $|r_{12}(\omega_{c})|=|r_{\sigma}(\omega_{c})|$, see Fig. 2(b). From Fig. 2, when $|r_{12}\left(\omega_{R,j}\right)|>|r_{\sigma}\left(\omega_{R,j}\right)|$, the changes of phase with frequency for these resonances are abnormally dispersive and the corresponding group delays are negative. When $|r_{12}\left(\omega_{R,j}\right)|<|r_{\sigma}\left(\omega_{R,j}\right)|$, the situation is totally reversed. For examples, in the case of $d=30\mu$m, all resonances are in the region of $|r_{12}\left(\omega_{R,j}\right)|>|r_{\sigma}\left(\omega_{R,j}\right)|$, therefore the group delays of the reflected light pulse are negative near resonances. In the case of $d=40\mu$m, one of resonances moves to the region of $|r_{12}|<|r_{\sigma}|$, thus the corresponding group delay becomes positive. As $d$ increases, there are more numbers of resonances moving into the region of $|r_{12}|<|r_{\sigma}|$; for instance, there are two resonances satisfying this condition in the plots for $d=90\mu$m. For a fixed graphene-based layered structure, $\varepsilon_{1}$, $\varepsilon_{2}$ and $d$ are usually fixed and cannot be changed. However, it is expected that we can change $r_{\sigma}$ by controlling the Fermi energy and temperature, thus the propagation of light reflection is automatically manipulated.

From the above discussion on Fig. 2, it is observed that the critical frequency $\omega_{c}$ at which the condition of $|r_{12}(\omega_{c})|=|r_{\sigma}(\omega_{c})|$ holds is very important for implementing the transition of phases and group delays near the corresponding resonances. Since $r_{12}$ is a constant, one can change $r_{\sigma}$ by adjusting the Fermi energy and temperature. In Fig. 3(a), we show that an increase in the value of $E_{F}$ greatly shifts the curve for $\left|r_{\sigma}\right|$ to the higher-frequency region. Thus the value of $\omega_{c}$ in the system increases linearly with $E_{F}$, see the inset in Fig 3(a). Therefore, the reflected group delays near the frequencies of resonances change their signs as $E_{F}$ increases in Fig. 3(b), and the light pulses at different carrier frequencies can be manipulated from negative to positive group delay reflection by simply adjusting the value $E_{F}$, see the inset in Fig. 3(b).

The effect of temperature on light reflection in such systems is very subtle because temperature only controls the transition part of conductivity $\sigma$ near the $2E_{F}$. From Fig. 1(c), it is seen that changing temperature has more distinct effect on the real part of $\sigma$, which represents the absorption of the graphene layer. Of course, there are significant changes in the imaginary part of $\sigma$ near the frequencies close to $2E_{F}$. In Fig. 4, we show the effect of the temperature $T$ on the light reflection in this system. Similar to the case of Fig. 3, the curve of $r_{\sigma}$ is shifted by changing $T$, but the critical value of $\omega_{c}$ is not linear with $T$. The dependence of the reflected group delays on $T$ is very sensitive when the resonance of the system happens around the frequency satisfying the equation $|r_{12}(\omega_{c})|=|r_{\sigma}(\omega_{c})|$, see Fig. 4(b). Another interesting feature is that, at the extremely low temperature, the group delay of the reflected light pulse shows the sharp change near $2E_{F}$.

Now we extend the above results into the cases of inclined incidence. When the angle of incidence is $\theta<\theta_{c}$, the above results can be readily obtained even for the cases of inclined incidence. Based on the above condition $|r_{12}(\omega_{c})|=|r_{\sigma}(\omega_{c})|$, we illustrate the dependence of the critical frequency $\omega_{c}$ on incident angle in Fig. 5. From Fig. 5, we see that the critical frequency $\omega_{c}$ increases before the Brewster angle $\theta_{B}$. Near $\theta_{B}$ there are two values of $\omega_{c}$ satisfying the above condition. This is not surprising since $r_{\sigma}$ is not a monotonic function of frequency. For certain angles, there are two regions satisfying $|r_{12}|<|r_{\sigma}|$. For example, as illustrated in Fig. 5, there may be only one point of intersection between $|r_{12}|$ and $|r_{\sigma}|$ for angles $\theta\in[0,33.413{{}^{\circ}})$ and $\theta\in(33.95{{}^{\circ}},\theta_{c})$ or two points of such intersection for angles $\theta\in(33.413{{}^{\circ}},33.63{{}^{\circ}})$ and $\theta\in(33.749{{}^{\circ}},33.95{{}^{\circ}})$, and even no point of such intersection for angles between $33.63{{}^{\circ}}$ and $33.749{{}^{\circ}}$. Therefore, by choosing the interesting frequency region and adjusting the Fermi energy and temperature, we can naturally manipulate the group delays in such systems.

Lastly, we discuss the case of total internal reflection (i.e., $\theta>\theta_{c}$). In this situation, the layered structure may be seen as an Otto structure. It is known that the effective dielectric constant of the graphene layer can be given by $\varepsilon_{\sigma}=1+i\sigma/\omega\varepsilon_{0}t_{g}$, where $t_{g}=0.5$nm is the thickness of graphene sheet Koppens2011 . Thus Re$\left[\varepsilon_{\sigma}\right]<0$ if Im$\left[\sigma\right]>\omega\varepsilon_{0}t_{g}$. In this sense, the graphene layer can be considered as the suitable alternative to a metal. It is well known that the surface plasmon can be excited at a metal-dielectric interface, and there is the counterintuitive dispersion effect in the Otto configuration with metals Wang2016 . When the graphene layer displays the properties of a metal, it is expected that the light reflection in graphene-based Otto structures may also be manipulated through the excitation of surface plasmon resonances.

In Fig. 6(a), we show the typical excitation of surface plasmon resonance (SPR) in such graphene-based Otto systems. There exists an optimal angle to excite the SPR for a fixed value of thickness $d$. For example, in the case of $d=300\mu$m, the optimal excitation of SPR is located at the angle around $63.93{{}^{\circ}}$ denoted by the vertical dashed line in Fig. 6(a). This optimal angle decreases along the dispersive curve of the ideal surface plasmon, see the inset of Fig. 6(a), as the value of $d$ increases. In Fig. 6(b), we show the physical properties of light reflection, its phase shift and group delay before and after the optimal angle of the SPR. It is clear that the pulse reflection suffers the transition from positive to negative group delay. Similarly, there exists an optimal thickness $d$ for a certain angle of incidence. As shown in Fig. 7(a), in the case of $\theta=60{{}^{\circ}}$, the optimal thickness $d$ is around 368$\mu$m. Near the excitation frequency of the SPR, it is clearly seen that the reflected group delay is tunable by adjusting the thickness $d$, see Fig. 7(b).

Furthermore, as discussed in the cases of resonance, the Fermi energy and the temperature may also significantly affect on the properties of light reflection in the cases of SPR excitations. Thus the reflected group delays are tunable by controlling the values $E_{F}$ and $T$. In Fig. 8(a), we show the linear relation of the SPR frequency vs the Fermi energy, and the group delays show different behaviors as $E_{F}$ changes. There also exists a transition value of $E_{F}$, which may lead to the changes of group delay from positive to negative values near the SPR frequency, see the right part of Fig. 8(a). It can be seen that the effect of $E_{F}$ on the SPR frequency is similar to the previously discussed resonant situations. However only one optimal $E_{F}$ exists when other parameters (like temperature, thickness $d$ and angle) are unchanged. There are similar effects of temperature $T$ on the SPR frequency and group delays. Of course, the effect of temperature on the SPR frequency is not linear, see Fig. 8(b).

IV Summary

We have presented the tunable transition effect of light pulse reflection in a layer system which contains a graphene layer. It is shown that there are two mechanisms to control the properties of light pulse reflection. When the angle of incidence is less than the critical angle of total reflection, the reflected group delay can be greatly tuned near the frequencies of resonances where the magnitudes of the Fresnel reflection coefficients satisfy certain conditions. In the case of total reflection, the optimal excitation of the SPR is the critical value to adjust the properties of pulse reflection in such Otto systems. It is demonstrated that the reflected group delays are controllable and tunable by adjusting the Fermi energy and temperature of the graphene sheet. The reflected group delays can also manipulate via changing the structural parameters including the position of the graphene layer and the angle of incidence. These results are useful to understand the fundamental physics underlying graphene-based light-matter interactions, and they may have potential applications in graphene-based optical signal processing and optical sensing.

Acknowledgements.

This work is supported by the National Natural Science Foundation of China (NSFC) (grants No. 11674284 and U1330203). It is also supported by the Fundamental Research Funds for the Center Universities (No. 2017FZA3005). This research is also supported by NPRP Grant No. 8-751-1-157 from the Qatar National Research Fund. L. Wang was supported by the China Scholarship Council (Grant No. 201606320146).