Applied Surface Science 453 (2018) 358–364
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Full Length Article
Broadly tunable and bidirectional terahertz graphene plasmonic switch based on enhanced Goos-Hänchen effect
T
⁎
Ali Farmani , Ali Mir, Zhaleh Sharifpour School of Electrical and Computer Engineering, Lorestan University, 465, Khoramabad, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Optical switch Graphene plasmonic Graphene surface conductivity White graphene Surface plasmon
A plethora of research in recent years has been reported on free space optical switches based on Goos-Hänchen shift of the reflected light in the surface plasmon resonant systems. However, very little research has reported the tunable Goos-Hänchen shift in a fixed configuration. Thus, the main purpose of this investigation is to: (a) Bidirectional and tunable switch consisting of graphene plasmonic Kretchmann configuration is designed based on the analytical approaches at the terahertz frequency range, (b) Owing to tunable optical properties of graphene, the effect of different parameters including the chemical potential, temperature, and scattering time of graphene on the Goos-Hänchen shift are investigated. Our analytical calculations show that by strong coupling the incident light to the surface plasmons of the structure, giant Goos-Hänchen shift as high as 540 times the free space wavelength can be achieved. Furthermore, the application of white graphene as the substrate of graphene layer increases the propagation of the graphene surface plasmons while the required external voltage decreases. It is also shown that by considering the small change of chemical potential: Δμc = 0.4 eV (external voltage of ΔV = 0.5 V), the Goos-Hänchen shift variation of 440 λ 0 (λ 0 = 1.55 μm ) can be easily provided. Finally, to verify our analytical results, the proposed structure is numerically simulated using finite-difference time-domain method. Based on these findings, this work present an alternate ways for improving both the tunability and magnitude of the Goos-Hänchen shift in a fixed configuration for bidirectional switching applications.
1. Introduction Switches as a optical devices have been caught in the spotlight of attention, particularly in the groundbreaking research including holography [1], modern medicine [2], terahertz telecom [3], and color switching [4]. From structural perspective, there are two categories of switches presented in the scientific literature as the free space configurations [5–7] and the guided wave configurations [8–10]. Typically, the underlying switching mechanism referees to the manipulation of the optical properties of the materials, in the structures [11]. The free space switches with two well-known configurations including Kretschmann and Otto configurations have gained plentiful attention during recent years [12–16]. The fundamental switching mechanism in free space switches is attributed to the lateral shift of an incident light which can be achieved by Newton’s optiks [17]. According to Newton’s optiks, when an incident light impinges at the boundary of the two different medium, a part of electromagnetic field penetrates into the rarer medium and makes an evanescent wave whose amplitude from both sides decreases exponentially [18]. In this case, the energy of the evanescent wave is transferred along the
⁎
interface; hence, after re-enter into the former medium, the reflected energy is laterally shifted with respect to the non-specular reflection [19,20]. This concept has been introduced and developed by F. Goos and H. Hänchen who in 1947 which is known as Goos-Hänchen (GH) shift [21,22]. Owing to notable characteristics of this effect, it is remarkably favorable in many other optical devices such as filters [23], sensors [24–26], thickness detector [27] and so on. Moreover, the GH shift has been studied extensively in various wavelength ranges including visible [28], infrared [29], microwave [30], and THz [31]. On the other hand, the top most desired features of a tunable switch are: low driving voltage, and tunability. To this purpose, some worthwhile effort has been made toward achieving switches with appropriate properties [32,33]. However, in some of these structures the magnitude of GH shift is smaller or comparable to the wavelength of the incident light. To enhance the tunability of switches, between miscellaneous platforms, plasmonic structures are good candidate [34]. Nobel metals with relatively good light-matter interaction are the appropriate candidate for presentation of plasmonic switches to be used for achieving high tunability [35]. Therefore, many authors have used plasmonic structures to achieve high tunability. Sui et al. reported a long-range
Corresponding author. E-mail addresses:
[email protected] (A. Farmani),
[email protected] (A. Mir),
[email protected] (Z. Sharifpour).
https://doi.org/10.1016/j.apsusc.2018.05.092 Received 21 March 2018; Accepted 13 May 2018 0169-4332/ © 2018 Elsevier B.V. All rights reserved.
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supports TM surface plasmons (i.e. waves with no magnetic field component in the plane of interface). From the other perspective, when the Fermi level is moved below the threshold point this structure supports TE modes (i.e. waves with no electric field component in the plane of interface). By considering strong coupling of the incident light to these waves, large GH of reflected light is realized. The numerical results extracted from the finite difference time domain analysis of presented structure is used in the Fresnel formula to calculate the reflection of incident light. The phase of reflection is then utilized to compute the GH shift for the different condition. Results presented here display that graphene together white graphene as a new smart material is a promising substitute for typical plasmonic switch, the GH shift can be harnessed with a minor variation in the Fermi level by applying small external voltage in the order of 1.6 V in turn leads to low power consumption. Using an analytical approaches, by strong coupling the incident light to the surface plasmons of the structure, giant Goos-Hänchen shift as high as 540 times the free space wavelength can be achieved. It is also shown that by considering the small change of chemical potential: Δμc = 0.4 eV (external voltage of ΔV = 0.5 V), the Goos-Hänchen shift variation of 440 λ 0 (λ 0 = 1.55 μm ) can be easily provided. The rest of this paper is organized as follows. In Section 2, the structure of graphene plasmonic switch is presented. Then, GH shift of the structure is calculated. To this purpose, the graphene dielectric constant is calculated and further investigated through analytical simulation. It is shown that by considering optical properties of graphene including chemical potential, temperature, and scattering time, the Fermi level of graphene can be harnessed and as a result of that the propagation of TM surface plasmons can be controlled. In the same Section white graphene is inspected and proposed as a superior substrate for the graphene. In Section 3, by utilizing the analytical and numerical method and considering strong coupling condition, we investigate the effect of different parameters including chemical potential, temperature and scattering time on GH shift. Then, briefly the main parameters of the proposed structure is compared with previous works. Finally, we summarize the main conclusions in Section 4.
surface plasmon configuration to realize large GH shift [36]. Moreover, Enhancement of lateral shift by surface plasmon excitation has also been studied by Li et al. [37]. The other related work came from Kar et al., who proposed a prism-waveguide to realize large GH shift [38]. Similarly large GH shift using the leaky mode has also been proposed by Matsunaga et al. [39]. Many of these structures act via three mechanism including changing the thickness of metal layer, modifying the susceptibility of the medium, and applying an external voltage. However, some of these platforms suffer from several inherent limitations, namely small light-matter interaction, high ohmic loss, and difficult tunability that restrict device performance as well as the mobility of these devices. To alleviate these issues, smart materials including phosphorene, silicene, and graphene are presented which can be improved the tunability of switches [40–45]. As far as we know, plentiful attention is given to graphene when it is the most promising candidate as tunability and strong light-matter interaction are important [46–51]. Graphene, an atom-thick bi-dimensional allotrope of carbon forming hexagonal structure with highly adjustable properties and strong light-matter interaction is introduced as a smart material which can be enhanced the performance of the switches. Besides, owing to its extraordinary properties such as its 0.023 absorption in the white light spectrum, high electronic mobility, low ohmic loss, high surface area, and adjustable energy bandgap, graphene has recently attracted great attention from researcher in both theoretical and experimental switching applications [52]. In a number of recent theoretical and empirical investigations, the GH shift in graphene-containing structure is inspected, which can support surface plasmons along the interface between graphene and dielectric [53,54]. Notwithstanding graphene plasmonic switches can be used as a appropriate platform, to improve the efficiency of these structures, many advanced materials such as white graphene as a graphene substrate are introduced [55–57]. White graphene has recently attracted a lot of attention because of its similarity to graphene in structure, and its exceptional thermal properties, and chemical stability. Although the structures of white graphene and graphene are similar, their electronic and optical properties are very different. While graphene is a semimetal (zero-gap), white graphene is a good insulator (or a wide bandgap semiconductor with Eg = 5.2–5.97 eV). The white graphene has been shown to be a superior substrate for graphene-based structures. The mobility values measured for graphene on the white graphene substrate are one order of magnitude higher than the graphene mobility fabricated on SiO2/Si substrate. Since white graphene is chemically and thermally stable, and free of dangling bonds and surface charge traps, it can show surface roughness and fluctuations two orders of magnitude lower than comparable substrates such as SiO2, which in turn leads to improved mobility, and chemical stability of a graphene layer when placed on the white graphene substrate [58]. Notwithstanding these valuable efforts, the giant tunability of lateral displacement in fixed configuration remains a problem. For the purpose of providing a giant and tunable GH shift in a fixed configuration, it is necessary to find out a new scheme to harnessing the graphene surface plasmons only with the external gate. In our previous works, we proposed Otto configuration to increasing the propagation of surface plasmons by harnessing the graphene surface conductivity in terahertz region [59]. Different from this work, we design a new tunable graphene plasmonic switches with modified Kretchmann configuration and use the finite difference time domain and analytical approaches to analysis the structure as well as investigate the effect of incident polarization, applied electric voltage, temperature, scattering time, and graphene substrate in near infrared region. In the present model, excitation of the surface plsmons on the graphene layer and therefore different switching condition can be achieved by harnessing the chemical potential of graphene by applying an external voltage. By applying an applied electric voltage the Fermi level can be easily shifted around the threshold point of; |μC | = ℏω/2 . As the Fermi level is moved above the threshold point this structure
2. Model and theory 2.1. The structure of the proposed switch A cross-sectional perspective of the proposed plasmonic switch with modified Kretschmann configuration together with multiple reflection of the incident light from this platform is illustrated in Fig. 1. In the present paper, the proposed structure consists of four layers of nonmagnetic (i.e., μ1 = μ 2 = μ3 = μ4 = 1) materials: dielectric prism, graphene, white graphene and silicon substrate. The dielectric prism has the refractive index of 3.5. The plasmonic region is provided by graphene/white graphene which is defined on a silicon substrate, aligned
Fig. 1. 2D-Schematic representation of the proposed structure for huge GH shift of TM surface plasmons. 359
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Fig. 4. The calculated GH shift of reflected light as a function of incident angle, as external voltage changes from 1 to 9 V.
Fig. 5. The calculated GH shift as a function of incident angle with various chemical potential (a): μC = 0.4 eV (green line), (b): μC = 0.6 eV (red line), and (c): μC = 0.8 eV (blue line). The inset shows the GH shift of single surface reflecting, as μC = 0.4 eV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. (a): The real part of graphene dielectric constant as a function of wavelength, as scattering time is; τ = 2 ps at T = 3° , (b): the imaginary part of graphene dielectric constant as a function of wavelength, as scattering time is; τ = 2 ps at T = 3° .
results. First, white graphene does not highly affect the carrier mobility of graphene, resulting in high-speed carrier mobility. Second, the roughness of white graphene compared to other substrate is so small that propagation of surface plasmons over long distances is feasible. Likewise, the introduction of silicon provides CMOS compatibility and enhances the capability of integration with silicon-based structures. Similarly, the dielectric constant of silicon in the interested wavelength is 3.44 and of thickness tSi = 300 nm. The prism region having a triangle shape, regarding low propagation loss, and the length of this region is attributed to maximum GH shift of the reflected light. The top of each graphene layer, external electric voltage is provided to inject the carriers into the graphene layer (see Fig. 1). The underlying principle of the structure is based on the plasma dispersion phenomenon, in which the modification of the dielectric constant of graphene is related to its carrier concentration. In this model, it is considered an unpolarized incident light with angular frequency of ω , under incident angle of θ impinges from dielectric prism (with a permittivity of ε4 ) into a graphene layer (with a permittivity of ε3 ). As can be seen, the proposed switch is composed of multiple reflecting graphene surfaces, aligned in a way that the reflected light of each surface illuminates the successive surface. Each reflecting surface supports a surface plasmons and strong coupling of the incident light to this surface waves results in large GH shift. The magnitude of the overall shift is obtained by sum of all shift values in reflection of the incident light from each surface, in the proposed modified Kretschmann configuration. It is assumed unpolarized 1.55 μ m (i.e., f = 198 THz) incident light with beam waist of α = 0.5 μm (i.e., α = 0.3λ ) will be launch into the
Fig. 3. The calculated phase of reflected light as a function of incident angle, as external voltage changes from 1 to 9 V.
in a way that the incident light excites the surface plasmons of the graphene. In this model, graphene is placed on white graphene layer
(
and its dielectric constant is calculated by εg = 1−
Im (σ ) ωt g
)+j
Re (σ ) , ωt g
where t g is the effective thickness of the graphene and is assumed to be t g= 0.3 nm. The calculation of dielectric constant of graphene through Kubo formula is obtained in the next subsection. The wavelength-dependent dielectric constant of white graphene is obtained from [5], and its thickness is similar to graphene layer. Thanks to the extraordinary properties of white graphene behind of its traditional counterparts such as SiO2, the application of white graphene leads to two remarkable 360
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Fig. 6. The calculated GH shift and reflected light as a function of incident angle. (a), (b): as τ = 2 ps, and T = 3° , (c), (d): as τ = 2 ps, and T = 300° , and as τ = 4 ps, and T = 3° .
Fig. 7. The field profile for TM GH shift of the proposed structure; (a): when T = 3° , (c): if T = 300° . The single GH shift of the structure is provided in (b), and (d).
the structure when the surface conductivity of the graphene is less (or higher) than that of threshold value. In the present model, harnessing the optical properties of the graphene including chemical potential, scattering time, and temperature can flexibly tune the lateral shift of the reflected light of the structure. In the next subsection, to show the tunable propagation of surface plasmons of graphene in turn tunes the GH shift of reflected light, the dielectric constant of graphene is calculated considering the Kubo
structure, bidirectionally, and therefore, with suitable graphene optical properties the surface plasmons at the interface of graphene/ white graphene can be excited. The graphene layers, hence, are biased with an external electric voltage attributed to the threshold point of the graphene chemical potential (2μc = ℏω) which above that the propagation polarization of the surface plasmons modifies from TE to TM. As a result, a graphene plasmonic switch is completed by tuning the applied electric voltage. The TE (or TM) incident light will propagate in 361
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Table 1 Comparison between the proposed structure and the previously reported research. Reference
λ 0 (μm)
Chenglong [60] Tahmasebi [61] Solookinejad [62] Yu [63] This Work
THz Regime 0.625 9.3 MW Range 1.55
GH/λ 0
Switching mechanism
Structure
200 64 450 >1 540
Harnessing Thickness and Voltage Altered Surface Roughness Dipole-dipole Interaction Magneto-optical Effect Tuned Graphene Optical Properties
Graphene Containing Structure Anisotropic Media Three level Nanoparticle Metamaterial Structure Graphene Plasmonic Structure
formula. Then, the analytical analysis is used in the Fresnel equations to calculate the phase of the totally reflected light. The calculated phase is then used to compute the GH shift according to the Artmann’s stationary phase approximate (SPA).
rTM = |rTM |exp (ϕ) = r123 +
The excitation of surface plasmons can be considered through the optical surface conductivity of graphene. The surface conductivity of graphene including interband and intraband transmissions can be expressed as in Eq. (1), in which the real and imaginary parts of the surface conductivity of graphene, respectively, represent the optical loss and stored energy. As we see in following equations, the surface conductivity of the graphene can be tuned by adjusting the temperature, scattering time (scattering rate), and also harnessing the chemical potential of the graphene around the threshold point of |μC | = ℏω/2 . For σg > 0 , the graphene layer shows metallic behavior and the structure supports transverse magnetic (TM) surface plasmons. For σg < 0 , the graphene layer displays dielectric behavior and the structure supports transverse electric (TE) surface plasmons.
S=−
σinter =
je 2
∞
π ℏ2 (ω−j2Γ)
∫0
(1)
je 2 (ω + j2Γ) π ℏ2
∫0
∞
ε(
∂fd (ε ) ∂ε
−
∂fd (−ε ) ∂ε
fd (−ε )−fd (ε ) ε
(ω−j2Γ)2−4( ℏ )2
) dε
(2)
dε (3)
τ −1
where ω is radian frequency, Γ = is scattering rate, e is the charge of the electron, κB = 1.38 × 10−23j/ k is the Boltzmann constant, ℏ is the reduced Planck constant, T is the temperature, μC is the graphene chemical potential, and fd (ε ) = (e ε − |μc |/ κB T + 1)−1 is the Fermi level. As mentioned before, in the present work TM surface plasmons is assumed, therefore, the interband terms of the graphene is so small compared with the intraband terms, so it can be negligible. By solving Eqs. (1)–(3), we can provide the expressions for the surface conductivity of the graphene as
σGraphene = σintra = −j
e 2κB T 2 π ℏ (ω−j2Γ)
μc ⎛ μc + 2ln ⎛e− κB T + 1⎞ ⎞ ⎝ ⎠⎠ ⎝ κB T
⎜
⎟
(4)
Likewise, the relationship between external electric field and chemical potential of graphene can be expressed as
Eext =
e επ ℏ2νF2
∫0
∞
fd (ε,μc )−fd (ε,2μc ) dε
(5)
here, ε is the dielectric constant of white graphene, and νF = 106 m/s is the Fermi velocity. To achieve electrical-dependent GH shift in the modified Kretchmann configuration, the reflection coefficient can be expressed as
r (θ,ω) = |r (θ,ω)|e jϕ (θ,ω)
λ 1 dIm|rTM | dRe|rTM | (Re [rTM ] −Im [rTM ] ) 2π |rTM |2 dθ dθ
(8)
Considering abovementioned equation, as the light is incident from a denser medium to a thinner medium, and the total reflection condition can be satisfied; therefore, the GH shift is occurred. In this case, maximum of the GH shift occurs close to the Brewster angle for a TM incident light owing to an abrupt change of phase of reflected light. In the following, we first briefly study the optical properties of white graphene. Then, the remarkable effect of the substrate on the propagation constant of the surface plasmons is investigated. As we know, the energy gap of graphene layer is notably affected by substrate. Why the white graphene is suitable as a substrate for graphene? Here are the answers to the question. When the graphene/substrate interaction is weak, many inherent properties of graphene such as mobility can be protected. As a result the external voltage required for inducing a specific change in the chemical potential of the graphene will be relatively low; μC = ℏνf απEext , where νf is Fermi velocity, and Eext is external voltage. White graphene has similar mobility with graphene and therefore, low interaction leads to low gating voltage: low power consumption. On the other word, the difference between the effective index of the surface plasmons and the refractive index of the surrounding dielectric media can be significantly increased by using white graphene as the graphene substrate. This mainly because of the higher effective surface conductivity of the graphene on white graphene structure compared to other conventional structures such as graphene on Si/SiO2. The constructive effect of the white graphene substrate in increasing the effective surface conductivity of the graphene/white graphene structure (or equivalently the effective carrier mobility) is usually attributed to the atomic-level smooth surface of the white graphene (rms roughness of white graphene is ∼50 pm, while the rms roughness of SiO2 is ∼250 pm), and absence of the “dangling” bonds or charge inhomogeneity at white graphene/graphene interface. In the next Section, first the dielectric constant of graphene is calculated. Then, we use analytical methods to calculate GH shift for TM modes in the proposed structure. To display the tunable behavior of GH shift, the effects of chemical potential, scattering time, and temperature are investigated. Likewise, to verify our analytical results, the numerical results through finite difference time domain is provided. Our both results display that when the incident angle is close to Brewster angle (strong coupling condition) the magnitude of GH shift can be much greater than incident wavelength owing to the excitation of surface plasmons.
where
σintra =
(7)
where the indices i = 1, 2, 3, 4 refer to the structure materials including the silicon, white graphene, graphene, and the dielectric prism layer, respectively, and t ij , rij , and rijk , are the Fresnel transmission coefficient, the Fresnel reflection coefficient of the shifted reflected light, and the Fresnel reflection coefficient of the specular reflected light from the three layers of i, j, and k, respectively. Having the Fresnel reflection coefficient of the non-specular reflected light, one can calculated value of GH shift by using of SPA method. Utilizing to this method for the light with beam waist around the incident wavelength, the GH shift can be analytically calculated as
2.2. The dispersion relation of the structure
σ(Graphene) = σintra + σinter
t13 t31 r34exp (2ik3 d2) 1−r34 r321exp (2ik3 d2)
(6)
where the reflection for TM surface plasmons considering boundary condition can be obtained from Fresnel equations as 362
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3. Results and discussion
According to Eqs. (1)–(4), the complex surface conductivity of graphene remarkably relays on scattering time; τ = μ × μc / e × νf2 , where μ = 10,000 cm2/(V s) is the dc mobility, μc is the chemical potential of graphene layer, and νf = 106 m s−1 is the Fermi velocity. As we know, the incident light can never be wholly coupled to surface plasmons owing to loss mechanism of graphene. Therefore, the GH shift is also depend on scattering time (loss mechanism). Here, as T = 3°, by changing of incident light angle from 40° to 80° and scattering time from τ = 2 ps to τ = 4 ps, the GH shift has been calculated through analytical method. In Fig. 6(e), (f), we plot the GH shift and reflected light as τ = 4 ps. This figure depicts the higher value of scattering time is, the smaller GH shift are near to Brewster angle. To give a deeper understanding and to verify the above analytical analysis, we perform the numerical simulations of GH shift based on finite difference time domain, where we assume a Gaussian beam incident from prism to graphene/white graphene interface. As shown in Fig. 7, as τ = 2 ps, temperature of graphene forcefully affects the GH shift; therefore, the GH shift can be harnessed by adjusting the temperature as well as chemical potential. In Fig. 7(a), (b), we plot the field profile for TM incident light for multiple and single reflected condition as T = 3°, and in Fig. 7(c), (d), the field profiles are shown, as T = 300°. This numerical result is in consonance with the analytical result for the GH shift in the modified Kretchmann structure. To further evaluate the structure, this model and obtained analytical and numerical results is compared against the previously reported works in Table 1. In brief, compared to the results and the structure of the references that have considered GH effect, here the value of the achievable GH shift is much larger and can be easily harnessed in a wide range with much smaller external voltage owing to use of white graphene. Finally, the proposed structure can be utilized as a tunable and bidirectional switch.
As mentioned in the Section 2, the external electric voltage on the graphene layer can be used to show the propagation of surface plasmons according to the Kubo formula. In Fig. 2, we have calculated and plotted the alternation of dielectric constant of the graphene when the external voltage changes from below the threshold value (i.e., |μC | < ℏω/2 ) to above the threshold value (i.e., |μC | > ℏω/2 ). As depicted in Fig. 2, when |μC | > ℏω/2 , the real part of surface conductivity is positive, in this case the intraband conductivity term becomes the dominant term in the dielectric constant of graphene supporting a TM surface waves (distinguished by green line in Fig. 2(a), (b)). On the other hand, as 0 < |μC | < ℏω/2 , the real part of dielectric constant is negative. In this case the interband conductivity is the dominant term and the it is shown in Fig. 2 will support TE surface waves (distinguished by blue region in Fig. 2(a), (b)). The GH shift can be enlarged by exciting the TM surface plasmons when the angle of the incident light is larger than the angle of the total internal reflection. To this purpose, the angle of incident light changes from 35 to 80 and phase of reflected light express by ϕr = Im (r )/ Re (r ) = Im (ln (r )) . As can be seen in Fig. 3, when the angle of incident light is larger than critical angle (i.e., θ = θsp = 58°), the phase of reflection abruptly changes from a positive to negative values, which leads to giant positive lateral displacement near resonance angle. Likewise, the sign of lateral displacement can be also expressed by the imaginary part and the real part of the reflected light. In such case the sign of lateral displacement is positive when magnitude of the Im (r ) is larger than the magnitude of Re (r ) . The analytical calculations show that the maximal GH shift can reach approximately 540 times that of the wavelength at the incidence angle θ = 58°, as displayed in Fig. 4. 3.1. The effect of chemical potential
4. Conclusion To investigate the effect of the chemical potential variation on the GH shift value, we have calculated this shift by using methods described in Section 2. Results of the calculation of GH are depicted in Fig. 5. According to this figure for the chemical potential values of; μc = 0.4 eV, μc = 0.6 eV, and μc = 0.8 eV, with the strong coupling between the incident light and surface plasmons of the structure, and therefore, the maximum shift value occur at the incident angle of; θ = 58°,θ = 60°, and θ = 61°, respectively. Moreover, GH shift as high as 540 λ 0 , 250 λ 0 , and 100 λ 0 , respectively, are achievable in the proposed structure. These very large shift values which are much larger than previously reported values, can be attributed to both the strong coupling between incident light and surface plasmons of the structure, and also the effect of white graphene as a graphene substrate. To highlight the effect of the multiple resonance together graphene substrate in the proposed Kretchmann structure on the GH shift improvement, GH shift of the reflected light from a single graphene Kretchmann structure as μc = 0.4 eV is also illustrated in the insets of Fig. 5.
In summary, a graphene plasmonic switch to improve the GH shift of reflected light in terahertz frequency based on analytical and numerical studies were investigated. By considering tunable optical properties of graphene, we have considered the effect of chemical potential, temperature, and scattering time of graphene plasmonic layer, to harness of GH shift close to Brewster angle. It was shown that, by considering strong coupling condition between incident light and surface plasmons of graphene, the magnitude GH shift can be two orders greater than an operating wavelength. Furthermore, the effect of the different parameters including the chemical potential, temperature, and the scattering time of the graphene layer, on the GH shift value was also studied. It was also shown that by introducing the small change of 0.4 eV, in the chemical potential of the graphene, the GH shift variation of 440 λ 0 could be achieved. We envision that this theoretical result leads to potential applications in graphene plasmonic integrated circuits such as bidirectional optical switches.
3.2. The effect of temperature and scattering time
Appendix A. Supplementary material
As presented in Section 2, and expressed in Eq. (4), it is clear that the magnitude of GH shift can be harnessed through modifying temperature of graphene layer. As can be seen in Fig. 6(a), (c), as τ = 2 ps, similar to the effect of increasing of graphene chemical potential through external voltage, by increasing temperature of graphene layer from T = 3° to T = 300°, the GH shift decrease, but temperature has a smaller effect on GH shift behind of chemical potential. In this case, the non-specular reflected light decreases by increasing temperature, as shown in Fig. 6(b), (d). In the experiment perspective, by increasing the temperature of graphene layer, the dielectric constant of prism slightly decreases which leads to lower propagation constant of surface plasmons; β . Consequently, the total group velocity increases; therefore, the GH shift decrease.
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