Electrically tunable Goos-H&#x00E4;nchen shifts in weakly absorbing epsilon-near-zero slab

Chenglong Wang; Faqiang Wang; Ruisheng Liang; Zhongchao Wei; Hongyun Meng; Hongguang Dong; Haifeng Cen; Ning Lin

doi:10.1364/OME.8.000718

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_f), 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.

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: $ε_{a i r} = 1, μ_{E N Z} = μ_{a i r} = 1, ε_{E N Z} = 0.001 + 10^{- 5} i$ . 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].

Fig. 1 (a) The schematic diagram of lateral shifts of transmitted and reflected beams from graphene-covered ENZ slab. (b) Multilayer graphene sheets surrounded by two different dielectrics that with dielectric constants ε₁ and ε₂, graphene sheets are characterized by conductivity σ' at z = 0. Arrows in different directions indicate incoming and outgoing light, respectively.(c) The calculation model for Fig. 1(a) used in transfer matrix method, in this structure we set ε₁ and ε₃ indicate the permittivity of air, d is the thickness of ENZ metamaterials.

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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:

E_{y 1} = A_{1} \cdot e^{- i k_{z 1} \cdot z} \cdot e^{i k_{x 1} \cdot x} + B_{1} \cdot e^{i k_{z 1} \cdot z} \cdot e^{i k_{x 1} \cdot x}, Z < 0

E_{y 2} = A_{2} \cdot e^{- i k_{z 2} \cdot z} \cdot e^{i k_{x 2} \cdot x} + B_{2} \cdot e^{i k_{z 2} \cdot z} \cdot e^{i k_{x 2} \cdot x}, Z > 0

Here $k_{z 1}$ and $k_{x 1}$ ( $k_{z 2}$ and $k_{x 2}$ ) 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 $k_{x 1} = k_{x 2}$ with Snell’s law. Then, we can apply boundary conditions at z = 0 and with Ohm’s law to get the transmission matrix $T_{1 \to 2, s}$ :

[\begin{matrix} A_{1} \\ B_{1} \end{matrix}] = T_{1 \to 2, s} [\begin{matrix} A_{2} \\ B_{2} \end{matrix}]

where

T_{1 \to 2, s} = [\begin{matrix} \frac{1}{2} + η_{s} + ζ_{s} & \frac{1}{2} - η_{s} + ζ_{s} \\ \frac{1}{2} - η_{s} - ζ_{s} & \frac{1}{2} + η_{s} - ζ_{s} \end{matrix}]

,

η_{s} = \frac{k_{z 2}}{2 k_{z 1}}, ζ_{s} = \frac{σ μ_{0} ω}{2 k_{z 1}}

.

$σ$ , $μ_{0}$ , $ω$ are surface conductivity of graphene, permeability of vacuum and frequency of incident beam respectively. For p polarization, we have:

[\begin{matrix} A_{1} \\ B_{1} \end{matrix}] = T_{1 \to 2, p} [\begin{matrix} A_{2} \\ B_{2} \end{matrix}]

where

T_{1 \to 2, p} = [\begin{matrix} \frac{1}{2} + η_{p} + ζ_{p} & \frac{1}{2} - η_{p} + ζ_{p} \\ \frac{1}{2} - η_{p} + ζ_{p} & \frac{1}{2} + η_{p} - ζ_{p} \end{matrix}]

,

η_{p} = \frac{ε_{1} k_{z 2}}{2 ε_{2} k_{z 1}}, ζ_{p} = \frac{σ k_{z 2}}{2 ε_{0} ε_{2} ω}

.

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:

P (d) = [\begin{matrix} e^{- i k_{z} \cdot d} & 0 \\ 0 & e^{i k_{z} \cdot d} \end{matrix}]

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

[\begin{matrix} A_{Ι} \\ B_{Ι} \end{matrix}] = T_{1 \to 2, s (p)} \cdot P (d) \cdot T_{2 \to 3, s (p)} [\begin{matrix} A_{Ι Ι Ι} \\ B_{Ι Ι Ι} \end{matrix}] = D [\begin{matrix} A_{Ι Ι Ι} \\ B_{Ι Ι Ι} \end{matrix}]

r = \frac{D_{21}}{D_{11}}, t = \frac{1}{D_{11}} .

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]:

σ (ω) = \frac{e^{2} E_{f}}{π ℏ^{2}} \frac{i}{ω + \frac{i}{τ}} + \frac{e^{2}}{4 ℏ^{2}} [θ (ℏ ω - 2 E_{f}) + \frac{i}{π} \log | \frac{ℏ ω - 2 E_{f}}{ℏ ω + 2 E_{f}} |]

in which E_f and τ are Fermi energy(chemical potential) and electron-phonon relaxation time.

τ = \frac{μ E_{f}}{e v_{f}^{2}}

, we choose the mobility μ = 10⁴cm²V⁻¹S⁻¹,

e

and

ℏ

are electron charge and reduced Plank’s constants respectively.

θ (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} = ℏ v_{f} \sqrt{π \cdot n_{2 D}}

, where v_f = 10⁶ m/s is the Fermi velocity of the graphene and

n_{2 D}

is the charge density, so we can control E_f through adjusting

n_{2 D}

controlled by an applied gate voltage, thereby leading to a voltage-controlled surface conductivity

σ (ω)

and G-H shifts .

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

r (d, θ, ω) = | r (d, θ, ω) | e^{i ϕ_{m}^{r} (d, θ, ω)}

t (d, θ, ω) = | t (d, θ, ω) | e^{i ϕ_{m}^{t} (d, θ, ω)}

where

ϕ_{m}^{r (t)} (d, θ, ω)

is the phase of r and t. And m represents s or p polarization. According to the stationary phase method, when incident beam with sufficiently large beam waist (i,e., the beam with a narrow angular spectrum

Δ k < < k_{0}

), the G-H shift can be calculated by [12]:

D_{m}^{r (t)} = - \frac{λ}{2 π} \frac{d ϕ_{m}^{r (t)}}{d θ}

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

D_{m}^{r (t)} = - \frac{λ}{2 π} \frac{1}{{| r {(t)}_{m} |}^{2}} [Re (r {(t)}_{m}) \frac{d Im (r {(t)}_{m})}{d θ} - Im (r {(t)}_{m}) \frac{d Re (r {(t)}_{m})}{d θ}]

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.

Fig. 2 G-H shift at different temperatures(s polarization). Here we set that E_f = 0.2ev, the thickness of the slab d/λ = 18.55 and τ = 100fs.

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The frequency of incident beam is 20Thz. It can be clearly seen from Fig. 3 that with the different values of ε₂, 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 ε₂ = 10⁻³ + 10⁻⁵i in this paper.

Fig. 3 The effect of different ε₂ on G-H shift in the case of reflection(s polarizatons). Here we set E_f = 0.2ev, N = 7 and τ = 100fs as well as θ = 1° in (a) and d/λ = 18.55 in (b).

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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.

Fig. 4 The effect of incident beam with different frequencies on G-H shifts (s polarization). (a) and (b) are reflected conditions, (c) and (d) are transmitted conditions, respectively. We choose the incident angle with θ = 1° in (a) and (c), then with the same thickness of ENZ metamaterial slab (d/λ = 18.55) in (b) and (d). The number of layers of graphene (N) is 7.

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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_zd = mπ(m = 1,2,3,.....), there are the maximal values of G-H shifts and while k_zd = 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 ε of ENZ metamaterials, a small k_z 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 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_f = 0.2ev, 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_f = 0.8ev, 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_f changes from 0.2ev to 0.8ev, we can achieve a wider range of adjustment. When we choose N = 7, G-H shifts have the largest adjustment range from about −250λ to −400λ. Therefor, one can increase the value of N and choose a suitable value of E_f according to one’s demand. The physical principle that E_f and N can change G-H shift is that the change of E_f 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.

Fig. 5 Reflected beam’s G-H shifts with s polarized incident beam in different conditions. (a) is the dependence of G-H shifts on the thickness of ENZ slab at different Fermi energy and the number of layers of graphene as well as the incident angle θ = 1°. The solid line and dash line represent E_f = 0.8ev and E_f = 0.2ev respectively, the different number of layers of graphene are marked with different colors. (b) is the dependence of G-H shifts on the angle of incident beam θ with the thickness of ENZ slab of d/λ = 18.55, other parameters are the same as (a). (c)describes the effect of different τ on G-H shift with E_f = 0.2ev and d/λ = 18.55. (d)The curve of G-H shifts as the function of Fermi energy, and we set that θ = 1°,the thickness of ENZ slab d/λ = 18.55 . Here we set that τ = 100fs in (a) (c) and (d), N represent the number of layers of graphene.

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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 0.8ev) lead to smaller negative G-H shifts than that of lower E_f (such as 0.2ev), 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: $τ = \frac{μ E_{f}}{e v_{f}^{2}}$ . So changing the value of τ for G-H shifts is just as effective as changing the value of E_f, and in next discussion we will not discuss the impact of τ on G-H shift. In order to make the relationship between G-H shift and Fermi energy(E_f) 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 E_f (see Fig. 5(d)) . It can be clearly seen that the value of G-H shift is large when the value of E_f is small. Even more interesting is that in Fig. 5(d), G-H shift changes from positive to negative as E_f changes, and we can realize the switching of G-H shift changes between positive and negative by tuning the Fermi energy.

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 200λ to 450λ) than that of reflected conditions.

Fig. 6 Transmitted beam’s G-H shifts with s polarized incident beam in different conditions. (a) is dependence of G-H shifts on the thickness of ENZ slab at different Fermi energy and layers of graphene as well as the incident angle θ = 1°.The solid line and dash line represent E_f = 0.8ev and E_f = 0.2ev respectively, different layers of graphenes are marked with different colors. (b) is the dependence of G-H shifts on the angle of incident beam θ with the thickness of ENZ slab of d/λ = 18.55, other parameters are the same as (a). (c)The curve of G-H shifts as the function of Fermi energy, with θ = 1°, the thickness of ENZ slab d/λ = 18.55 . Here we also set that τ = 100fs.

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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_f is, the larger the adjustment range can be obtained.

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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Electrically tunable Goos-Hänchen shifts in weakly absorbing epsilon-near-zero slab

Abstract

1. Introduction

2. Model and method

2.1 Transfer matrix method

2.2 G-H shift

3. Results and discussions

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (12)

Optical Materials Express