## Abstract

We theoretically calculated the Goos-Hänchen (G-H) shift of the beam that reflected from and transmitted through an epsilon-near-zero (ENZ) slab, which was covered by the different number of layers of graphene and also realized tunable G-H shifts with electrically controllable graphene in terahertz regime. It is shown that besides the impact of the thickness of the slab and the number of layers of graphene, Fermi energy (chemical potential),which can be electrically controlled through electrical modification of the charge density of graphene by gate voltage, also plays an important role in adjusting G-H shifts. In this work we achieved about 200 times the incident wavelength of the adjustment range which can be used in measuring the doping level of graphene due to the dependence of Fermi energy on G-H shifts. Furthermore, our results provide a richer control on G-H shifts in ENZ slab and also provide potential applications for ENZ metamaterials-based devices than semi-infinite structures.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The graphene, as a type of single atomic sheet of graphite, has attracted scientists’ interest due to its special physical properties [1,2]. The surface plasmons in graphene shows stronger confinement of plasmonic excitation than regular metals, which means that graphene can be a new platform to research light-matter interactions [3]. Graphene also shows great potential applications in nano-devices fields because of its ultrahigh electron mobility and ultrafast relaxation time [4,5]. What’s more, we can also use these properties of graphene to achieve dynamic control especially in terahertz and far-infrared regime [6,7]. Science the optical properties of graphene can be adjusted by varying the Fermi level (*E _{f}*) through electrostatic gating or chemical doping [8], there are also some articles that utilize these properties of graphene to discuss dynamically adjustable sensors [9,10] etc. And these articles provide the possibility of realizing dynamic manipulation of light by changing the optical properties of graphene.

Goos-Hänchen (G-H) effect, a tiny lateral shift of reflected light beam, was first discovered in 1947 [11], then it was explained with stationary phase method [12] and energy propagation method [13]. Because of its interesting properties, G-H effect in many aspects has important applications especially in modern science, such as high sensitivity solution concentration measurement [14], temperature measurement [15], displacement detector [16] and other weak measurements [17] fields. Hence, G-H effect has received extensively attentions. In past years, G-H effect has been researched from traditional medium to new artificial material [18], photonic crystals [19], and other absorptive dielectrics [20,21]. Recently, with the advent of new materials, there are new features with G-H effect in epsilon-near-zero (ENZ) metamaterials and a few researchers have concerned G-H shifts from ENZ metamaterials. Ya et al. [22] calculated G-H shift in semi-infinite ENZ metamaterials and found that G-H shift tends to zero with p-polarized incident light and tends to a constant value while in s-polarized incident light. Zia et al. [23] researched G-H shift in semi-infinite ENZ metamaterials with partially coherent light, and discussed the impact of beam width, spatial coherence and different mode of light on G-H effect. Grosche et al. [24] theoretically studied the G-H shift of a Gaussian beam on the surface coated with one layer of graphene and showed how conductivity of graphene affects the G-H shift. Furthermore, Yuan et al. [25] achieved the dynamic control of G-H shifts by covering one layer of graphene on semi-infinite ENZ metamaterials and Farmani et al. [26] also discussed in detail the effect of *E _{f,}* temperature and the waist radius of beam on it. In addition, some papers also discuss G-H effect in graphene double-barrier structures and quantum G-H effect in graphene [27,28]. However, the thickness of media also plays an important role in G-H shifts, and in fact, the dielectric has a certain thickness. Due to the highly demand of G-H effect in modern applications, it is very meaningful to achieve dynamic adjustment of G-H effect especially in dielectric slab.

In this letter, we achieve large and adjustable G-H shifts while light is reflected from and transmitted through the graphene-covered slab. We discussed in details the impact of Fermi level (*E _{f}*) and the number of layers of graphene (N) on the G-H effect. It’s easy to turn the G-H shift by changing Fermi energy (

*E*), we also find that with the different number of layers of graphene, the scope of dynamic adjustment is not the same, so we can choose the right N according to our demand. Furthermore, with this structure, we can get much larger G-H shifts than that of the previous structure [25,29], which has promise application in the field of high sensitivity sensors.

_{f}## 2. Model and method

We illustrate our structure in Fig. 1(a), d is the thickness of ENZ metamaterials and the surface of ENZ metamaterial is coated with multiple layers of graphene. In the case of few number of layers, in order to simplify the calculation, there is an approximate relationship σ' = Nσ [30](see Fig. 1(b)). Therefore, we take multi-layer graphene equivalent to one layer of graphene that with the conductivity of σ'. What’s more, for convenience compare the results to previous literature [29], we set same parameters: ${\epsilon}_{air}=1,{\mu}_{ENZ}={\mu}_{air}=1,{\epsilon}_{ENZ}=0.001+{10}^{-5}i$. Because the light beam is a collection of plane wave of slight different transverse wave vectors, then the G-H shift can be calculated with stationary-phase method [12]. As for the phase of reflection and transmission coefficient, we can calculate it with the transfer matrix method [31,32].

#### 2.1 Transfer matrix method

Considering the propagation of light in a homogeneous medium, the s polarized incident light, which electric field is polarized along y direction can be written as:

Here ${k}_{z1}$and ${k}_{x1}$(${k}_{z2}$and${k}_{x2}$) are the wave-vectors of z and x component in media 1 and media 2, respectively,${A}_{1(2)}$and${B}_{1(2)}$are field coefficients. It’s also easy to get that ${k}_{x1}={k}_{x2}$with Snell’s law. Then, we can apply boundary conditions at z = 0 and with Ohm’s law to get the transmission matrix ${T}_{1\to 2,s}$:

$\sigma $,${\mu}_{0}$,$\omega $are surface conductivity of graphene, permeability of vacuum and frequency of incident beam respectively. For p polarization, we have:

According to the idea of transfer matrix method, we can use a similar approach to calculate the electric field at z = d and get${T}_{2\to 3,s(p)}$. Note that while light is propagating through the ENZ metamaterials, the incident light on the lower surface(z = d) has an additional phase. So we calculate this additional phase with a propagation matrix:

Then one can get the whole transfer matrix as well as coefficients r and t:

#### 2.2 G-H shift

Within the random-phase approximation (Note that Eq. (6) should be used at zero temperature, but we can also employ a finite-temperature extension [33].), the surface conductivity of graphene in local limit reduces to [34]:

*E*and τ are Fermi energy(chemical potential) and electron-phonon relaxation time.$\tau =\frac{\mu {E}_{f}}{e{v}_{f}^{2}}$, we choose the mobility μ = 10

_{f}^{4}cm

^{2}V

^{−1}S

^{−1},$e$and$\hslash $are electron charge and reduced Plank’s constants respectively.$\theta (x)$is the Heaviside step function. The first term in Eq. (6) corresponds to intraband contribution and the second term corresponds to interband contribution. What’s more,${E}_{f}=\hslash {v}_{f}\sqrt{\pi \cdot {n}_{2D}}$, where

*v*10

_{f}=^{6}m/s is the Fermi velocity of the graphene and${n}_{2D}$is the charge density, so we can control

*E*through adjusting${n}_{2D}$controlled by an applied gate voltage, thereby leading to a voltage-controlled surface conductivity$\sigma (\omega )$ and G-H shifts .

_{f}From Eqs. (5), the coefficients r and t can also be written as the from:

While in absorption materials, G-H shifts should be expressed by [21]:

Then with Eq. (10), we can analyze the impact of graphene on G-H shifts in ENZ slab.

## 3. Results and discussions

In this paper, we only consider the situation of zero temperature, in order to further illustrate the reliability of our result, we also discussed the impact of temperature on the G-H shift. As shown in Fig. 2, we calculated the G-H shift at two different temperatures (T = 0K and T = 300K). Obviously there is not much difference between G-H shifts at two different temperatures, so it is reasonable for us to set temperature as 0K here.

The frequency of incident beam is 20Thz. It can be clearly seen from Fig. 3 that with the different values of ε_{2}, the change of G-H shift is very obvious. In the case of the same real part, the smaller the imaginary part is, the larger the G-H shift is. On the contrary, in the case of the same imaginary part, the larger the real part is, the larger the G-H shift is (We can see that the value of the point on the blue line is almost zero). So we choose ε_{2} = 10^{−3} + 10^{−5}i in this paper.

Then, we considered the impact of frequencies on G-H shifts shown in Fig. 4. It can be clearly seen from Fig. 4 that the influence of incident waves of different frequencies on G-H shifts. Note that lower frequency, such as 20Thz, could provide lager adjustment range of G-H shifts than that of higher frequency, such as 200Thz. So we choose 20Thz as the incident frequency in next discussion.

We considered the incident beam with s polarization and the reflected beam’s G-H shifts which are shown in Fig. 5. It’s easy to see that G-H shifts changes cyclically as the distance *d* changes in Fig. 5(a). This is because while *k _{z}d* = mπ(m = 1,2,3,.....), there are the maximal values of G-H shifts and while

*k*= mπ-π/2 there are the minimal values of G-H shifts from Eq. (9) and we should use Eq. (10) while calculating the expressions of G-H shifts with lossy dielectric. Due to the very small value

_{z}d*ε*of ENZ metamaterials, a small

*k*could lead to a large oscillatory period. What’s more, lager thickness of slab lead to smaller G-H shifts which has been shown in Fig. 5(a). To see more clearly we have drawn a part of Fig. 5(a) in the upper right corner with d/λ in 18~20. With the incident angle

_{z}*θ*= 1°, lower Fermi energy can achieves lager negative G-H shifts than that of higher Fermi energy while with same number of layers of graphene. Furthermore, the number of layers of graphene also plays an important role in changing G-H shifts. For lower Fermi energy such as

*E*= 0.2

_{f}*ev,*which marked by dash lines in Fig. 5(a), different N almost has no effect on G-H shifts (around d/λ = 19). But on the contrary, for higher Fermi energy such as

*E*= 0.8

_{f}*ev,*which marked by solid lines, with the bigger value of N, the smaller G-H shifts can be achieved. So with the more number of layers of graphene, while the

*E*changes from 0.2

_{f}*ev*to 0.8

*ev,*we can achieve a wider range of adjustment. When we choose N = 7, G-H shifts have the largest adjustment range from about −250

*λ*to −400

*λ.*Therefor, one can increase the value of N and choose a suitable value of

*E*according to one’s demand. The physical principle that

_{f}*E*and N can change G-H shift is that the change of

_{f}*E*or N leads to the change of the conductivity of graphene, thus changing the boundary conditions of the electromagnetic field and finally changing the G-H shift.

_{f}In addition, for the incident light of s-polarization state in reference [25], the G-H shift of the reflected light is almost zero when the incident angle is small. However, while considering the medium thickness, we can get a large G-H shift. And in addition to the Fermi energy, changing the number of layers of graphene can also properly adjust the range of G-H shift, which will be helpful for practical engineering applications.

In Fig. 5(b), the dependence of G-H shifts on incident angle at different Fermi energy and different value of N with *d/λ* = 18.55 are also discussed. G-H shifts have the largest absolute value near *θ* = 1°(about 400 times the incident wavelength), which is larger than previous. Furthermore, with the same value of N, higher *E _{f}* (such as 0.8

*ev*) lead to smaller negative G-H shifts than that of lower

*E*(such as 0.2

_{f}*ev*), which is similar to Fig. 5(a). Then with N = 7, G-H shifts also have the largest adjustment range than other values of N. In addition, it can be seen from Eq. (6) that relaxation time

*τ*can also affect the conductivity of graphene and the G-H shift, so we investigate the effect of

*τ*on G-H shift in Fig. 5(c). However, Fig. 5(b) and Fig. 5(c) are very similar due to that Fermi energy and

*τ*have the following relationship: $\tau =\frac{\mu {E}_{f}}{e{v}_{f}^{2}}$. So changing the value of

*τ*for G-H shifts is just as effective as changing the value of

*E*, and in next discussion we will not discuss the impact of

_{f}*τ*on G-H shift. In order to make the relationship between G-H shift and Fermi energy(

*E*) has a more intuitive image, we also make the curve of G-H shift as the function of Fermi energy to show that the adjustment of G-H shift with

_{f}*E*(see Fig. 5(d)) . It can be clearly seen that the value of G-H shift is large when the value of

_{f}*E*is small. Even more interesting is that in Fig. 5(d), G-H shift changes from positive to negative as

_{f}*E*changes, and we can realize the switching of G-H shift changes between positive and negative by tuning the Fermi energy.

_{f}For transmitted conditions, G-H shifts are always positive, this is because that transmitted phase always changes continuously whatever the loss is present or not. In addition that the transmitted G-H shifts are large positive due to the significant phase change near resonances. To see more clearly we have drawn a part of Fig. 6(a) in the upper right corner and discussed the impact of graphene on G-H shifts. In fact, with the increase of the thickness *d,* we can achieve lager G-H shifts (see Fig. 6(a)). Transmitted G-H shifts have nearly the same changes with reflected conditions, with the increase of value N we can get larger G-H shifts and the largest adjustment range can also be achieved with N = 7. The difference is that transmitted G-H shifts have larger adjustment range (from about 200*λ* to 450*λ*) than that of reflected conditions.

While determining the thickness of the slab *d* and with the same value of N, the Fermi energy also plays an important role, higher Fermi energy can provides larger G-H shifts than that of lower Fermi energy shown in Fig. 6(b). Then as the same as reflected conditions, multilayer graphene lead to lager G-H shifts and lager adjustment range which more than 400 and 200 times wavelength of incident beam respectively. Figure 6(c) shows the relationship of G-H shift with *E _{f}* , and G-H shift is always positive which is different from Fig. 5(d). The larger the value difference of

*E*is, the larger the adjustment range can be obtained.

_{f}Furthermore, we also analyzed the G-H shifts with p polarized incident beam and find that has a similar phenomenon of change with s polarized conditions, so we do not repeat the description with p polarized incident beam.

## 4. Conclusion

In summary, we proposed a new method to achieve controllable G-H shifts in terahertz regime by using the properties of graphene, which conductivity can be adjusting through gate voltage. We also theoretically studied the impact of different parameters on reflected and transmitted beam’s G-H shifts and find that in the case of lower Fermi energy with different layers of graphene, it has a smaller adjustment range than that of higher Fermi energy. What’s more, with the increase of value N(but we do not change the value of E* _{f}*), we can achieve larger adjustment range, which can be allowed to serve as a strategy to measuring the doping level(chemical potential or Fermi energy) of graphene. The electrical tunability of G-H shifts from graphene-covered ENZ slab could potentially open a new possibility of dynamic measurement and flexible optical-beam steering, etc.

## Funding

National Natural Science Foundation of China (NSFC) (61774062, 61275059, 11674109).

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