Quadruple plasmon-induced transparency of polarization desensitization caused by the Boltzmann function

Xiao Zhang; Fengqi Zhou; Fengqi Zhou; Zhimin Liu; Zhimin Liu; Zhimin Liu; Zhenbin Zhang; Yipeng Qin; Shanshan Zhuo; Xin Luo; Enduo Gao; Hongjian Li

doi:10.1364/OE.433258

1. Introduction

Over the last years, surface plasmon polaritons (SPPs), collective resonance of surface particle on the stimulation of incident electromagnetic wave, have received extensive attention [1–3]. A number of researchers have done some work by utilizing SPPs [4–8], owing to their excellent optical properties in respect of energy localization, information transfer, and polarization control. Moreover, some SPP-based devices have been applied to practice, such as absorbers, modulations, and polarizers [9–11]. However, the fact that SPPs are easily excited by linearly polarized light demonstrates that the SPP-based devices are highly affected by the angle of incident polarized light [12–14]. Based on this fact, the application of polarization sensitivity or insensitivity is proposed by many reports, wherein the polarization sensitivity property increases the freedom of modulation; the polarization desensitization results in the fact that some SPP-based devices are not affected by the polarized state of light source. As a typical case of SPPs, plasmon-induced transparency (PIT) [15,16] is also inseparable from the limitation of polarization direction of light source [17]. Its physical origin can be summarized as follows: the structures of materials can be resonant at some frequencies when linearly polarized light is incident, which is called bright modes; other structures regarded as dark modes will produce no obvious resonance in this polarized light source; the effect that dark mode suppresses resonance absorption of bright mode generates a transparent window [18–20]. The above condition of PIT generation needs a single linearly polarized plan wave. Hence, the angle of polarized light is also an important point to be considered for the PIT-based device. Many studies have reported the response of PIT phenomenon to polarized states of incident light [21–23]. New discoveries are taken to researchers by both the disappearance of bright mode effect and the appearance of dark mode effect that are being studied and attributed to the polarization sensitivity. However, the mutual conversion between bright and dark modes is not as simple as complementary excitation. It contains complex mathematical formulas and SPP excitation mechanism, which need to be further explored. It means that a more accurate mathematical description is needed to build the response mechanism model.

In addition, to explain the PIT phenomenon, researchers have explored the, hybrid theory, quantum level theory, and coupled mode theory (CMT). The hybrid theory is usually utilized to explore spectral coupling [24]; the quantum level theory generally takes the interaction between structures and incident light as energy states for the analysis of its microscopic characteristics [25–27]. Each theory has its own merits, but CMT is more suitable as a tool to be employed to demonstrate the PIT phenomenon and to fit with simulation [28,29]. The terahertz light refers to the electromagnetic wave from 0.1 THz to 10 THz. To regulate and control this light wave, we must realize the multiple plasmon-induced transparency in limited wave band, which requires that CMT must be applicable for multi-PIT phenomenon. Therefore, CMT has been deduced and calculated in the case of two or three modes, or even four modes [30,31]. However, in the case of multi-PIT, the increase of each resonant mode means that we should not only repeat the calculation of low dimension CMT for which the previous work has been done, but also explore high dimension CMT corresponding to the number of resonant modes. Therefore, it becomes particularly necessary to deduce n-order CMT formula, and obtain the general conclusion for this system.

This study proposes a periodic graphene metamaterial comprising a graphene block, four graphene strips, and a range of graphene arrangement composed of nine squares to realize dynamically tunable quadruple PIT on which the angle of polarized light has almost no effect. The Boltzmann function satisfied by the transmission response of graphene strips at resonant frequency to the angle of linearly polarized light was firstly proposed, which results in conjugated variety between graphene structures with varying polarization state of light. The structural centrosymmetry, and the conjugated variety that is caused by the combination of polarization sensitivity result in polarization desensitization of the proposed metamaterial. In addition, when n-order CMT is deduced and calculated, its eventual results in the case of n = 5 show close agreement with finite-difference time-domain simulations. As a result, a quintuple-mode on-to-off modulation based on simultaneous electro-optical switch is realized at different Fermi levels of graphene. Its modulation degrees of amplitude at 1.703 THz, 2.991 THz, 4.527 THz, 6.160 THz, and 7.142 THz are respectively 68.0%, 94.2%, 86.5%, 72.4%, and 92.0%, of which the dephasing time is respectively 7.1 ps, 3.2 ps, 5.0 ps, 6.7 ps, and 3.9 ps. Finally, the slow-light property based on n-order CMT is analyzed, and the n-order group index can reach 321 corresponding to n = 5 in the metamaterial. Therefore, to study multi PIT-based devices, the polarization desensitization due to a response that satisfies the Boltzmann function and the results of n-order CMT are first proposed.

2. Structural design and analysis

As shown in Fig. 1(a), a dielectric silicon-substrate with thickness of d = 300 nm is covered by a series of single-layer graphene arrays; its graphene structures comprising a graphene block, four graphene strips, and a range of graphene arrangement composed of nine squares. The periodic structural unit with a size of a = b = 3800 nm is adopted in x-y plane to form the proposed metamaterials. The top view of a structural unit is plotted in Fig. 1(b), with geometric parameters as follows: s₁ = s₂ = 1200 nm, s₃ = s₄ = l₁ = l₃ = 400 nm, l₂ = l₄ = 2200 nm. The schematic diagram of structural unit illuminated by incident light is shown in Fig. 1(c). Its expression for the light source is right circularly polarized light, which means this structure is desensitized to polarized light. Before the multi-polarization analysis, the linearly x-polarized light is incident and used as a light source for the analysis of the formation mechanism of quadruple PIT. The intensity of incident light is weak, and the nonlinear effect of graphene has little effect, so it is ignored. In addition, the biasing source and lines are necessary for tuning the Fermi level within graphene, which introduce some side effects to the original design in practice. However, the dynamic modulation of Fermi level is simulation calculation in this paper. Thus, the side effects by biasing line is ignored by idealization, here, the biasing source and lines are ignored.

Fig. 1. (a) Schematic diagram of the preparation process of the proposed metamaterials. (b) Top view of the structural unit. (c) Diagram of irradiated structural unit with the Fermi of 1.0 eV within graphene.

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The graphene structures are on the same layer, but their interactions with linearly polarized light exhibit asynchronism. For a better understanding, we divide the single-layer structure into several parts, as shown in Fig. 2(a). When a linearly x-polarized plane wave is normally transmitted along the z direction, quadruple PIT emerges, and its transmission spectra are illustrated in Fig. 2(b). Although the PIT phenomenon is normal compared with the double or triple PIT mentioned in the previous work, there are still non-simultaneous interactions. In Fig. 2(c), the graphene arrangement-squares (GASs) produce an effect of bright mode at 3.414 THz, as shown by the green Lorenz line; moreover, another bright mode is generated by the graphene block (GB) at 6.374 THz, as represented by the orange curve. The two structures can act as dark modes for each other owing to the different resonant frequencies. When GASs and GB exist at the same time, the blue-dotted line is thereby produced. Interestingly, only one bright mode is stimulated herein by the interaction; GB is excited into a transparent window, while GASs remain unchanged. The fact that a bright mode is changed into PIT by a dark mode can be seen from the electric field distribution. As shown in Figs. 2(i) and 2(j), at 6.374 THz, GB localizes energy before interaction, and then its field is dimmed after interaction. It indicates that single PIT emerges because the GB bright mode is stimulated by the effect of dark mode within GASs. However, at 3.414 THz, the field of GASs doesn’t change before or after interaction, as shown in Figs. 2(g) and 2(h), which means GASs retain the effect of bright mode. From the above two bright modes, it can also be seen that the PIT phenomenon is not so simple as a superposition of spectral lines. Another interesting fact is that the effect of extra bright mode is generated at 1.976 THz owing to the interaction between GASs and GB, which can be observed from the comparison between Figs. 2(e) and 2(f). It’s the reason why GASs are used as a structure, and the emergence of the extra bright mode is beneficial to the production of quadruple PIT. Generally, two bright modes should lead to the single PIT effect with only one transparent window or two transmission dips. However, in the process that they act as dark or bright modes for each other, the generation of extra bright mode at f₁, and invariance of bight mode GASs at f₂ result in two extra dips. In addition, the graphene horizontal strips (GHSs) and the graphene vertical strips (GVSs) act as the bright mode and the dark mode, respectively, as shown in Fig. 2(d). Therefor, the three remaining bright modes produce three transparent windows with quadruple PIT emerging subsequently. Furthermore, the different structure is excited at different resonant frequencies, as shown in Figs. 2(k)-2(o), which can be regarded as the five resonators of quadruple PIT.

Fig. 2. (a) Breakdown schematic diagram of asynchronous interaction of graphene structures in the same layer. (b) Transmission spectrum of the quadruple-PIT. (c-d) Transmission spectra of interaction among different graphene structures. (e-o) Electric field distribution at some frequencies. (p-s) Electric field distribution at the frequencies of four transmission peaks.

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3. Deduction of n-order CMT and calculation in the case of n = 5

To simplify calculations and summarize results, n-order CMT is deduced, and the results are discussed with n = 5 as an example. Here, the expression of “n-order” does not refer to the Laser mode or Electric field mode but it represents the number of resonators in CMT. As shown in Fig. 3(a), ‘M_n (n = 1, 2, 3, …)’ are imaginary resonators corresponding to n-order CMT with the amplitude a_n; its superscript ‘in/out’ and subscript ‘+/-’ of ‘Mn_+/-^in/out’ are respectively the input or output waves and positive or negative direction of wave propagation. It should be noted that any resonant dip is not just a single action of a graphene structure, and each structure works more or less, which can be seen from excited Electric field of several structures at resonant frequency. Thus, the corresponding relationship between the graphene structures and the imaginary resonators is not rigorous. But we need to know the fact that the quality factors of the imaginary resonators have important impacts to the five resonant dips of CMT. Before deduction of CMT, it’s necessary to obtain some parameters of the imaginary resonators in the first place. In the Kubo formula, the expression for the conductivity of graphene σ is [32,33]:

(1)$$\begin{aligned}\sigma &=\frac{{{e^2}(\omega + i{\tau ^{ - 1}})}}{{i\pi {\hbar ^2}}}[\frac{1}{{{{(\omega + i{\tau ^{ - 1}})}^2}}}\int_0^\infty {\varepsilon (\frac{{\partial F(\varepsilon )}}{{\partial \varepsilon }} - \frac{{\partial F( - \varepsilon )}}{{\partial \varepsilon }})} d\varepsilon - \int_0^\infty {\frac{{F( - \varepsilon ) - F(\varepsilon )}}{{{{(\omega + i{\tau ^{ - 1}})}^2} - 4{{(\varepsilon /\hbar )}^2}}}} d\varepsilon ]\\ &= {\sigma ^{{\mathop{\rm int}} ra}} + {\sigma ^{{\mathop{\rm int}} er}}\end{aligned}, $$

where, F(ɛ)={1+exp[(ɛ-µ_c)/ K_BT]} ^-1 is the Fermi Dirac distribution while τ=µE_f /eV_F² is the carrier relaxation time [34]. In the two formulas above, ɛ, ω, ħ, µ_c, µ, E_f, V_F, and T are respectively the dielectric of graphene, the angular frequency of the incident light, the reduced Planck constant, the chemical potential, the mobility of graphene, the Fermi level, the Fermi velocity, and the temperature of system. σ^intra is intra-band electron photon scattering while σ^inter is direct inter-band photon transitions [35]. However, σ^inter is ignored owing to the condition K_B<<2E_f. Therefore, the conductivity of the single-layer graphene can be obtained as [36]:

(2)$$\sigma = \frac{{i{e^2}{E_f}}}{{\pi {\hbar ^2}({\omega + i{\tau^{ - 1}}} )}}. $$

Fig. 3. (a) Schematic diagram of coupled mode theory.

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Based on the different media above and below the graphene layer, the propagation constant β [37] of graphene is calculated by:

(3)$$\frac{{{\varepsilon _{Si}}}}{{\sqrt {{\beta ^2} - {\varepsilon _{Si}}k_0^2} }} + \frac{{{\varepsilon _{air}}}}{{\sqrt {{\beta ^2} - {\varepsilon _{air}}k_0^2} }} ={-} \frac{{i\sigma }}{{\omega {\varepsilon _0}}}, $$

where, ɛ_Si, ɛ_air, ɛ₀, η₀, and k₀, are respectively the relative permittivity of silicon, the relative permittivity of air, the vacuum permittivity, the inherent impedance, and the wave vector. Hence, the effective refractive index is expressed as: n_e=β/k₀. Thereafter, the inter-loss quality factor Q_in and inter-loss coefficient γ_in are calculated respectively by Q_in=Re(n_e)/ Im(n_e) and γ_in = ω_n/2Q_in with ω_n as the nth resonant angular frequency. In addition, the total-loss quality factor Q_tn is obtained by Q_tn = f /Δf. Finally, the extra-loss coefficient γ_on is calculated by 1/ Q_tn = 1/ Q_in +1/ Q_on [38–40].

For n-order CMT, based on the conservation of energy, the outgoing wave and incoming wave for two adjacent resonators are as follows:

(4)$$M_{\textrm{n + }}^{in} = M_{(n - 1) + }^{out} \cdot {e^{i{\varphi _{n - 1}}}}, (n = 2,3,4, \ldots ), $$

(5)$$M_{\textrm{(n - 1) - }}^{in} = M_{n - }^{out} \cdot {e^{i{\varphi _{n - 1}}}}, ({\rm n} = 2,3,4, \ldots ), $$

(6)$$M_{\textrm{n + }}^{out} = M_{n + }^{in} - \gamma _{on}^{{\textrm{1} / \textrm{2}}} \cdot {a_n}, ({\rm n} = 2,3,4, \ldots ), $$

(7)$$M_{\textrm{n - }}^{out} = M_{n - }^{in} - \gamma _{on}^{{\textrm{1} / \textrm{2}}} \cdot {a_n}, ({\rm n} = 2,3,4, \ldots ), $$

where, the phase differences between adjacent resonators are obtained by φ_n = Re(β)d_n. According to the above relationships, the energy relationship of each resonator is expanded as follows:

(8)$$M_{\textrm{1 + }}^{in} \ne \textrm{0}, $$

(9)$$\begin{aligned} &M_{\textrm{1 - }}^{in} ={-} \gamma _{o2}^{{\textrm{1} / \textrm{2}}} \cdot {a_2} \cdot {e^{i{\varphi _1}}} - \gamma _{o3}^{{\textrm{1} / \textrm{2}}} \cdot {a_3} \cdot {e^{i({\varphi _1} + {\varphi _2})}}\\ &- \cdots - \gamma _{on}^{{\textrm{1} / \textrm{2}}} \cdot {a_n} \cdot {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - 1}})}} \end{aligned}, $$

(10)$$M_{\textrm{2 + }}^{in} = M_{\textrm{1 + }}^{in} \cdot {e^{i{\varphi _1}}} - \gamma _{o1}^{{\textrm{1} / \textrm{2}}} \cdot {a_1} \cdot {e^{i{\varphi _1}}}$$

(11)$$\begin{aligned} &M_{\textrm{2 - }}^{in} ={-} \gamma _{o\textrm{3}}^{{\textrm{1} / \textrm{2}}} \cdot {a_\textrm{3}} \cdot {e^{i{\varphi _\textrm{2}}}} - \gamma _{o\textrm{4}}^{{\textrm{1} / \textrm{2}}} \cdot {a_\textrm{4}} \cdot {e^{i({\varphi _\textrm{2}} + {\varphi _\textrm{3}})}}\\ &- \cdots - \gamma _{on}^{{\textrm{1} / \textrm{2}}} \cdot {a_n} \cdot {e^{i({\varphi _\textrm{2}} + {\varphi _\textrm{3}} + \cdots + {\varphi _{n - 1}})}} \end{aligned}, $$

(12)$$\begin{aligned} &M_{\textrm{(n - 1) + }}^{in} = M_{\textrm{1 + }}^{in} \cdot {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - 2}})}} - \gamma _{o1}^{{\textrm{1} / \textrm{2}}} \cdot {a_1} \cdot {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - 2}})}}\\ &- \gamma _{o2}^{{\textrm{1} / \textrm{2}}} \cdot {a_2} \cdot {e^{i({\varphi _2} + {\varphi _3} + \cdots + {\varphi _{n - 2}})}} - \cdots - \gamma _{o(n - 2)}^{{\textrm{1} / \textrm{2}}} \cdot {a_{n - 2}} \cdot {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - 2}})}} \end{aligned}, $$

(13)$$M_{\textrm{(n - 1) - }}^{in} ={-} \gamma _{on}^{{\textrm{1} / \textrm{2}}} \cdot {a_n} \cdot {e^{i{\varphi _{n - 1}}}}, $$

(14)$$\begin{aligned} &M_{\textrm{n + }}^{in} = M_{\textrm{1 + }}^{in} \cdot {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - \textrm{1}}})}} - \gamma _{o1}^{{\textrm{1} / \textrm{2}}} \cdot {a_1} \cdot {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - \textrm{1}}})}}\\ &- \gamma _{o2}^{{\textrm{1} / \textrm{2}}} \cdot {a_2} \cdot {e^{i({\varphi _2} + {\varphi _3} + \cdots + {\varphi _{n - \textrm{1}}})}} - \cdots - \gamma _{o(n - \textrm{1})}^{{\textrm{1} / \textrm{2}}} \cdot {a_{n - \textrm{1}}} \cdot {e^{i{\varphi _{n - \textrm{1}}}}} \end{aligned}, $$

(15)$$M_{\textrm{n - }}^{in} = \textrm{0}. $$

The first item within Eq. (8) hereinabove is kept in the other formulas because it will be eliminated at the final step. It is worthy of note that the input wave of negative direction is zero for final nth resonator. In addition, the transfer relationships of nth resonators can be expressed as:

(16)$$\left( {\begin{array}{cccc} {{\gamma_1}}&{ - i{\mu_{12}}}& \cdots &{ - i{\mu_{1n}}}\\ { - i{\mu_{21}}}&{{\gamma_2}}& \cdots &{ - i{\mu_{2n}}}\\ \vdots & \vdots & \ddots & \vdots \\ { - i{\mu_{n1}}}&{ - i{\mu_{n2}}}& \cdots &{{\gamma_n}} \end{array}} \right) \cdot \left( {\begin{array}{c} {{a_\textrm{1}}}\\ {{a_\textrm{2}}}\\ \vdots \\ {{a_n}} \end{array}} \right) = \left( {\begin{array}{cccc} { - \gamma_{o1}^{{1 / 2}}}&0& \cdots &0\\ 0&{ - \gamma_{o2}^{{1 / 2}}}& \cdots &0\\ \vdots & \vdots & \ddots & \vdots \\ 0&0& \cdots &{ - \gamma_{on}^{{1 / 2}}} \end{array}} \right) \cdot \left( {\begin{array}{c} {M_{1 + }^{in} + M_{1 - }^{in}}\\ {M_{2 + }^{in} + M_{2 - }^{in}}\\ \vdots \\ {M_{n + }^{in} + M_{n - }^{in}} \end{array}} \right), $$

where, γ_n=iω-iω_n-γ_in-γ_on (n=1, 2, 3, …), and µ_nm(m ≠ n) are the mutual coupling coefficients. When Eq. (16) is submitted with Eqs. (8–15), it is calculated as:

(17)$$\left( {\begin{array}{c} {\gamma_{o1}^{{\textrm{1} / \textrm{2}}} \cdot M_{1 + }^{in}}\\ {\gamma_{o2}^{{\textrm{1} / \textrm{2}}} \cdot M_{1 + }^{in} \cdot {e^{i{\varphi_1}}}}\\ \vdots \\ {\gamma_{on}^{{\textrm{1} / \textrm{2}}} \cdot M_{1 + }^{in} \cdot {e^{i({\varphi_1} + {\varphi_2} + \cdots + {\varphi_{n - 1}})}}} \end{array}} \right) = \left( {\begin{array}{c} \begin{array}{l} - {\gamma_1}{a_1} + (\gamma_{o1}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{o2}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i{\varphi_1}}} + i{\mu_{12}}){a_2} + (\gamma_{o1}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{o3}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i({\varphi_1} + {\varphi_2})}} + i{\mu_{13}}){a_3}\\ + \cdots + (\gamma_{o1}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{on}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i({\varphi_1} + {\varphi_2} + \cdots + {\varphi_{n - 1}})}} + i{\mu_{1n}}){a_n} \end{array}\\ \begin{array}{l} (\gamma_{o2}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{o1}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i{\varphi_1}}} + i{\mu_{21}}){a_1} - {\gamma_2}{a_2} + (\gamma_{o2}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{o3}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i{\varphi_2}}} + i{\mu_{23}}){a_3}\\ + \cdots + (\gamma_{o2}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{on}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i({\varphi_2} + \cdots + {\varphi_{n - 1}})}} + i{\mu_{2n}}){a_n} \end{array}\\ \vdots \\ \begin{array}{l} (\gamma_{on}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{o1}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i({\varphi_1} + {\varphi_2} + \cdots + {\varphi_{n - 1}})}} + i{\mu_{n1}}){a_1} + (\gamma_{on}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{o2}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i({\varphi_2} + {\varphi_3} + \cdots + {\varphi_{n - 1}})}} + i{\mu_{n2}}){a_2}\\ + (\gamma_{on}^{{\textrm{1} / \textrm{2}}} \cdot \gamma_{o3}^{{\textrm{1} / \textrm{2}}} \cdot {e^{i({\varphi_3} + {\varphi_4} + \cdots + {\varphi_{n - 1}})}} + i{\mu_{n3}}){a_3} + \cdots - {\gamma_n}{a_n} \end{array} \end{array}} \right), $$

and then ‘M₁₊ⁱⁿ’ is divided on both sides of Eq. (17). Moreover, given that the right part of this equation is a matrix with n rows and one column, it can be regarded as a product of an n×n matrix and an n×1 matrix. Therefore, Eq. (17) can be simplified as:

(18)$$\left( {\begin{array}{c} {\gamma_{o1}^{{1 / 2}}}\\ {\gamma_{o\textrm{2}}^{{1 / 2}} \cdot {e^{i{\varphi_1}}}}\\ \vdots \\ {\gamma_{on}^{{1 / 2}} \cdot {e^{i({\varphi_1} + {\varphi_2} + \cdots + {\varphi_{n - 1}})}}} \end{array}} \right) = \left( {\begin{array}{cccc} { - {\gamma_1}}&{{\kappa_{12}}}& \cdots &{{\kappa_{1n}}}\\ {{\kappa_{21}}}&{ - {\gamma_2}}& \cdots &{{\kappa_{2n}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{\kappa_{n1}}}&{{\kappa_{n2}}}& \cdots &{ - {\gamma_n}} \end{array}} \right) \cdot \left( {\begin{array}{c} {{\zeta_1}}\\ {{\zeta_2}}\\ \vdots \\ {{\zeta_n}} \end{array}} \right). $$

The fungible relationships herein are ζ_n = a_n/ M₁₊ⁱⁿ, and κ_pq = γ_op¹^/2γ_oq¹^/2exp(φ)+iµ_pq (p≠q; p, q = 1, 2, 3, …, n). Finally, the transmission and reflection coefficients are calculated and expressed as follows:

(19)$$\begin{aligned} &{t_n} = \frac{{M_{\textrm{n + }}^{out}}}{{M_{\textrm{1 + }}^{in}}} = {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - \textrm{1}}})}} - \gamma _{o1}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _1} \cdot {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - \textrm{1}}})}}\\ &- \gamma _{o2}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _2} \cdot {e^{i({\varphi _2} + {\varphi _3} + \cdots + {\varphi _{n - \textrm{1}}})}} - \cdots - \gamma _{o(n - \textrm{1})}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _{n - \textrm{1}}} \cdot {e^{i{\varphi _{n - \textrm{1}}}}} - \gamma _{on}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _n} \end{aligned}, $$

(20)$$\begin{aligned} &{r_n} = \frac{{M_{\textrm{1 - }}^{out}}}{{M_{\textrm{1 + }}^{in}}} ={-} \gamma _{o1}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _1} - \gamma _{o2}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _2} \cdot {e^{i{\varphi _1}}} - \gamma _{o3}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _3} \cdot {e^{i({\varphi _1} + {\varphi _2})}}\\ &- \cdots - \gamma _{on}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _n} \cdot {e^{i({\varphi _1} + {\varphi _2} + \cdots + {\varphi _{n - 1}})}} \end{aligned}. $$

The general rule of n-order CMT has been shown in Table 1. Therefore, the transmission and reflection can be are respectively obtained by T_n=| t_n |2, and R_n=| r_n |2, wherein the parameters ζ₁-ζ_n are obtained by solution of the Eq. (18). Since the absorption is equal to the normalized total energy minus the sum of the transmission and reflection, it expressed as A_n = 1- T_n - R_n. When n = 5 is taken as an example, the transmission and reflection coefficient is expressed as:

(21)$$\begin{aligned} &{t_\textrm{5}} = \frac{{M_{\textrm{5 + }}^{out}}}{{M_{\textrm{1 + }}^{in}}} = {e^{i({\varphi _1} + {\varphi _2} + {\varphi _\textrm{3}} + {\varphi _\textrm{4}})}} - \gamma _{o1}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _1} \cdot {e^{i({\varphi _1} + {\varphi _2} + {\varphi _\textrm{3}} + {\varphi _\textrm{4}})}}\\ &- \gamma _{o2}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _2} \cdot {e^{i({\varphi _2} + {\varphi _3}\textrm{ + }{\varphi _\textrm{4}})}} - \gamma _{o\textrm{3}}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _\textrm{3}} \cdot {e^{i({\varphi _3}\textrm{ + }{\varphi _\textrm{4}})}} - \gamma _{o\textrm{4}}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _\textrm{4}} \cdot {e^{i{\varphi _\textrm{4}}}} - \gamma _{o5}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _5} \end{aligned}, $$

(22)$$\begin{aligned} &{r_\textrm{5}} = \frac{{M_{\textrm{1 - }}^{out}}}{{M_{\textrm{1 + }}^{in}}} ={-} \gamma _{o1}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _1} - \gamma _{o2}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _2} \cdot {e^{i{\varphi _1}}} - \gamma _{o3}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _3} \cdot {e^{i({\varphi _1} + {\varphi _2})}}\\ &- \gamma _{o\textrm{4}}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _\textrm{4}} \cdot {e^{i({\varphi _1} + {\varphi _2}\textrm{ + }{\varphi _3})}} - \gamma _{o5}^{{\textrm{1} / \textrm{2}}} \cdot {\zeta _5} \cdot {e^{i({\varphi _1} + {\varphi _2}\textrm{ + }{\varphi _3}\textrm{ + }{\varphi _4})}} \end{aligned}, $$

the function relationship satisfied by the coefficients ζ₁, ζ₂, ζ₃, ζ₄, and ζ₅ is expressed as:

(23)$$\left( {\begin{array}{c} {\gamma_{o1}^{{1 / 2}}}\\ {\gamma_{o\textrm{2}}^{{1 / 2}} \cdot {e^{i{\varphi_1}}}}\\ {\gamma_{o3}^{{1 / 2}} \cdot {e^{i({\varphi_1} + {\varphi_2})}}}\\ \begin{array}{l} \gamma_{o4}^{{1 / 2}} \cdot {e^{i({\varphi_1} + {\varphi_2} + {\varphi_3})}}\\ \gamma_{o5}^{{1 / 2}} \cdot {e^{i({\varphi_1} + {\varphi_2} + {\varphi_3} + {\varphi_4})}} \end{array} \end{array}} \right) = \left( {\begin{array}{ccccc} { - {\gamma_1}}&{{\kappa_{12}}}&{{\kappa_{1\textrm{3}}}}&{{\kappa_{1\textrm{4}}}}&{{\kappa_{1\textrm{5}}}}\\ {{\kappa_{\textrm{21}}}}&{ - {\gamma_2}}&{{\kappa_{\textrm{23}}}}&{{\kappa_{\textrm{24}}}}&{{\kappa_{\textrm{25}}}}\\ {{\kappa_{\textrm{31}}}}&{{\kappa_{\textrm{32}}}}&{ - {\gamma_3}}&{{\kappa_{\textrm{34}}}}&{{\kappa_{\textrm{35}}}}\\ {{\kappa_{\textrm{41}}}}&{{\kappa_{\textrm{42}}}}&{{\kappa_{\textrm{43}}}}&{ - {\gamma_4}}&{{\kappa_{\textrm{45}}}}\\ {{\kappa_{\textrm{51}}}}&{{\kappa_{\textrm{52}}}}&{{\kappa_{\textrm{53}}}}&{{\kappa_{\textrm{54}}}}&{ - {\gamma_5}} \end{array}} \right) \cdot \left( {\begin{array}{c} {{\zeta_1}}\\ {{\zeta_2}}\\ {{\zeta_3}}\\ \begin{array}{l} {\zeta_4}\\ {\zeta_5} \end{array} \end{array}} \right). $$

Table 1. General rule of n-order CMT

View Table

4. Results discussion and application

E_f, a parameter in CMT, affects the final result dynamically. At different Fermi levels of graphene, the transmission and reflection exhibit great difference. The Fermi level within graphene is modulated as follows [41]:

(24)$${E_f} = \hbar {V_F}\sqrt {\frac{{\pi {\varepsilon _0}{\varepsilon _{Si}}{V_g}}}{{de}}}, $$

where, V_g is the gate voltage. Therefore, the results calculated by CMT at different Fermi levels are illustrated in Fig. 4, as shown by red-dotted curves. Additionally, the blue solid curves represent the simulation results by FDTD at different Fermi levels. The results fit well with each other, indicating that the n-order CMT is feasible in the case of n = 5 to fit with the FDTD simulation. The three-dimensional evolutions of transmission, reflection, and absorption are respectively illustrated in Figs. 4(b), 4(d), and 4(f). At the resonant frequencies, as the Fermi level increases, the transmissions become weaker, while the reflections ger stronger, which can be observed from the widened resonant region, such as the blue area of transmission and the red area of reflection. However, the absorption near the resonant frequencies is weakened. It is related to the metallic properties of the graphene for which the Fermi levels are set higher. These results agree with those in our previous research. Therefore, n-order CMT in this work can be extended to high dimensional CMT computation, and its results are rational.

Fig. 4. Transmission (a), reflection (c), and absorption (e) spectra obtained by the FDTD simulation and the CMT calculation at different Fermi levels. Three-dimensional evolutions of transmission (b), reflection (d), and absorption (f) spectra at different Fermi levels.

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To find the response of this metamaterial to the polarized angle, different graphene structures are illuminated in the case of different polarized light, as shown in Figs. 5(a)-5(d). Here, x-direction is the case that the angle of polarized light is 0°, and the condition that θ = 90° represents the y-polarized light. The red-dotted curves represent that a linearly y-polarized plane wave is incident, with the blue solid lines at x-polarization direction. At x-polarization, GVSs act as a dark mode, while GHSs act as a bright mode; however, at y-polarization, the dark/bright effects of GVSs and GHSs are interchangeable with each other. It is evident that their functions are opposite. The responses of the resonant dips generated by GVSs and GHSs to the angle of polarized light are respectively plotted in Figs. 5(e) and 5(f) with the form of polar coordinates; their radii herein represent the transmission, while the angle θ of polar coordinates corresponds to the angle of polarized light. According to the polar coordinates, the response of GVSs is that a dark mode changes into a bright mode with an increasing polarized angle from 0° to 90° by 5° every time. However, the change of GHSs is a converse process relative to the variety of the GVSs, which can be further observed from the three-dimensional evolutionary diagrams illustrated in Figs. 5(i) and 5(j). Here, the data which the evolution diagrams in Figs. 5(i)-5(l) need are processed by interpolation owing to its enormous data size. It implies that there is a process of mutual transformation between the two graphene structures with the change of the polarized angle. It should be noted that the blue regions in Figs. 5(i)-5(l) are not corresponding to the blue regions of Fig. 6(c) because they are the resonant regions of bright modes. The response of graphene structure to the polarization angle herein has been studied by some researchers, and was called ‘complementary excitation’ or ‘synchronization conversions’ [17,21,22], but it is actually a conjugate change according to our numerical calculation.

Fig. 5. (a-d) Transmission spectra of different graphene structures by mutually perpendicular polarized light. (e-h) Transmission of the resonant dips generated by different graphene structures, adopted as functions of θ. (i-l) Three-dimensional evolutionary diagram of different graphene structures at different directions of polarized light.

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Fig. 6. (a) The conjugated variety of resonant dip with the change of polarization direction. (b) The changing quadruple PIT adopted as functions of θ. (c) Three-dimensional evolutionary diagram of the quadruple PIT.

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The transmission responses of GHSs and GVSs to the angle of polarized light at resonant frequency 4.474 THz are shown respectively as red squares and blue triangles in Fig. 6(a), which is nonlinear. We try to fit and predict these response relationships as shown by green and light red curves. According to the calculation, they satisfy the Boltzmann function as follows:

(25)$$T = \frac{{p - q}}{{1 + {e^{({\theta - {\theta_0}} )/d\theta }}}} + q. $$

where, p, q, and θ₀ are fitting constants, whose values depend on the geometric parameters of graphene strips. For GHSs, p_h = 0.02056, q_h = 1.0292, θ_0h = 45.05131, while for GVHs, p_v = 1.0292, q_v = 0.02056, θ_0v = 44.94869. When θ = 45°, the values of TGHSs and TGVSs are equal with each other. When θ = 90°, TGHSs ≈ 1, and TGVSs ≈ 0. It is evident that the two processes of mutual inverse conversion are synchronous owing to the same geometric parameters in simulation. Moreover, the sum of TGHSs and TGVSs is a constant whatever the value of θ is. Hence, the responses of the GHS and GVS structures to the angle of polarized light both satisfy the Boltzmann function, resulting in the conjugate change. Based on the excitation law of SPPs, we speculate that this general Boltzmann law satisfied by the response to polarization states can be extended to other graphene or metal strips. In addition to GHSs and GVSs, GASs and GB, the other two graphene structures, are desensitized to the polarized angle owing to structural centrosymmetry, which can be seen from the circular curves in Figs. 5(g)-5(h) and the unchanged blue region in Figs. 5(k)-5(l). In fact, the centrosymmetry herein is resulted from the Boltzmann law of structural edge. The quadruple PIT with varied angle of polarized light is shown in Fig. 6(b), where the invariance of the PIT is obvious. Therefore, the evolution of the entire quadruple PIT illustrated in Fig. 6(c) demonstrates that the five resonant regions are unchanged despite the increasing polarized angle.

The phenomenon of red shift with a decreasing Fermi level can be clearly seen from the three-dimensional evolutionary diagram in Fig. 4(b), which presents another result about dynamic change of Fermi levels. On the one hand, the resonance absorptions of graphene structures result in the transmission dips; on the other hand, the lower Fermi levels of graphene can weaken the resonance effect via the occurrence of red shift. In addition, the resonant dip will shift to a low frequency and eventually disappear as the Fermi level decreases. Hence, the application of electro-optical switch is feasible. It must be pointed out that these results can be generalized to the case of incident light with any polarized angle owing to the structural conjugated variety and centrosymmetry, as shown in Fig. 7(a). The electro-optical switch desensitized to the polarized light is thereby realized as shown in Fig. 7(b), where its switch effect is quintuple-mode on-to-off simultaneous modulation with the working wavelength of 4×10^4 ∼ 3×10^5 nm approximately. When the Fermi level are set to 1.1 eV, the electro-optical switches corresponding to an “off” state respectively at 1.703 THz, 2.991 THz, 4.527 THz, 6.160 THz, and 7.142 THz, and change to an “on” state with the Fermi level set to 0.1 eV. Hence, the modulation degree of amplitude (MDA) is calculated by MDA = | A_on - A_off | / A_on×100% to describe the adjustable capability, where the MDAs of these switches are respectively 68.0%, 94.2%, 86.5%, 72.4%, and 92.0%. The insertion losses for them are respectively 12.2%, 10.5%, 5.8%, 4.9%, and 5.6%, here, it is simply expressed as the light opacity in “on” state of electro-optics switch, thus, we use the definition of “1-T_on”. Furthermore, considering that the sharpness of switch is prominently important for determination of device performance, the dephasing times, which are calculated by T = 2ħ / FWHM [42–44], are thus respectively estimated as 7.1 ps, 3.2 ps, 5.0 ps, 6.7 ps, and 3.9 ps. For further analysis of the more complex light sources, the RCP light composed of vertically linearly polarized light with a phase difference of π/2 and -π/2 is used as the light source to illuminate the structure, as shown in Fig. 7(a). And we find that the Quadruple PIT is unchanged.

Fig. 7. (a) Schematic diagram of the proposed metamaterials at right-circularly polarized light. (b) Schematic diagram of optical switch via comparision between transmission spectra at different Fermi levels. (c-f) Group index and phase shift at different Fermi levels.

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Finally, the slow-light properties corresponding to n-order CMT, an important application of PIT, are hereby analyzed. The group index n_g is an important parameter to define the effect of slow light. Its high value causes outstanding slow light. The relationship between the group velocity and the group index is n_g = c / ν_g, where the group velocity ν_g, a physical quantity to describe the speed of wave packet, is calculated by: ν_g = dω / dk. Therefore, the n-order group index is expressed as [45]:

(26)$${n_g} = \frac{c}{{{v_g}}} = c\frac{{dk}}{{d\omega }} = \frac{c}{h}\frac{{d{\phi _n}}}{{d\omega }}, $$

where, c, k, ω, and h are respectively the velocity of light in vacuum, the wave vector, the angular frequency, and the thickness of substrate, with ϕ_n as the transmission phase shift of n-order CMT system. It is obtained by ϕ = arg(t_n) in the case of n = 5 based on the proposed graphene metamaterials. As shown in Figs. 7(c)-7(f), the group index (red curves) and the phase shift (blue curves) at different Fermi levels are illustrated hereby. When the Fermi level of graphene is 1.1 eV, the group index of n = 5 can reach 321 with sharp change of the transmission phase shift. Therefore, the n-order group index is critical for the design of slow-light device for which the n-order CMT system is used.

5. Conclusion

In this work, we proposed a graphene metamaterial comprising a graphene block, four graphene strips, and a range of graphene arrangement consisting of nine squares to realize dynamically tunable quadruple PIT. Owing to the structural centrosymmetry, and the conjugated variety between the graphene structures, the quadruple PIT was desensitized to the angle of incident polarized light. In addition, we theoretically deduced n-order CMT, of which the results in the case of n = 5 agreed with the finite-difference time-domain simulations of the proposed metamaterials. As a result, a quintuple-mode on-to-off modulation based on simultaneous electro-optical switch was realized; the slow-light device based on n-order CMT was thereby proposed. In the case of n = 5, MDAs of the electro-optical switch at 1.703 THz, 2.991 THz, 4.527 THz, 6.160 THz, and 7.142 THz are respectively 68.0%, 94.2%, 86.5%, 72.4%, and 92.0%, of which the dephasing time is respectively 7.1 ps, 3.2 ps, 5.0 ps, 6.7 ps, and 3.9 ps; and the group index can reach 321. Therefore, the results of the n-order CMT, and the insights gained into the polarization-desensitization structure provide new research progress and references for the design of novel optoelectronic devices.

Funding

National Natural Science Foundation of China (11804093, 11847026, 12164018, 61764005); Natural Science Foundation of Jiangxi Province (20192BAB212003, 20202ACBL212005, 20202BABL201019); Graduate Innovative Special Fund Projects of Jiangxi Province (YC2020–S312).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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PIT phenomenon	The number of PIT	The number of couple modes	Transmission coefficient	Reflection coefficient
PIT phenomenon	m	n	tn	rn
Single-PIT	1	2	$\frac{M_{2 +}^{o u t}}{M_{1 +}^{i n}}$
Duble-PIT	2	3	$\frac{M_{3 +}^{o u t}}{M_{1 +}^{i n}}$
Triple-PIT	3	4	$\frac{M_{4 +}^{o u t}}{M_{1 +}^{i n}}$	$\frac{M_{1 -}^{o u t}}{M_{1 +}^{i n}}$
Quadruple-PIT	4	5	$\frac{M_{5 +}^{o u t}}{M_{1 +}^{i n}}$
…	…	…	…
Multi-PIT	k	k+1	$\frac{M_{(k + 1) +}^{o u t}}{M_{1 +}^{i n}}$

Quadruple plasmon-induced transparency of polarization desensitization caused by the Boltzmann function

Abstract

1. Introduction

2. Structural design and analysis

3. Deduction of n-order CMT and calculation in the case of n = 5

4. Results discussion and application

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (1)

Equations (26)

Optics Express