1. Introduction

Electromagnetically induced transparency (EIT) is a quantum interference effect that removes light absorption in an atomic media [1]. The phenomenon permits for a spectrally incommodious optical transmission window accompanied with extreme dispersion, which is greatly desirable for sensing, filter and slow light applications [2–4]. Recent studies have showed that EIT-like optical resonances can be acquired in plasmonic metamaterial systems at optical-radio frequencies [5–17]. Especially, since the idea of a plasmonic EIT analogue (plasmon induced transparency: PIT) has initially notionally been proposed [5], EIT-like resonances grounded on a dipole-quadrupole coupling [5, 8–10], that occur intense spectral features of the quadrupole resonance and therefore demonstrate excessive hypersensitivity to structural or environmental changes, are playing a progressively remarkable role in physics and applications. One of the most crucial PIT nanostructures consists of three nanorods, in which one nanorod is parallel to the polarized direction of the incident light and the other two nanorods are perpendicular to it [9, 18-19]. In these PIT nanostructures, the quadrupolar modes of vertical nanorods (dark mode) are excited by the dipole modes of parallel nanorods (bright mode), resulting in a PIT in the spectrum. For the most part of these PIT nanostructures, this single dipole-quadrupole coupling induces a single wavelength PIT effect. On the other side, metamaterial systems encouraging multiple coupled dipole-quadrupole are able to provide multiple PIT resonances windows [20], which will disclose a significant channel toward metamaterial applications operating at multiple frequency fields. In general, the multi-spectral PIT effects are generated with multiple coupling in intricate layered or stacked structures [20, 21].

However, the little modulation depth in the resonances, as well as the manufacture of the sophisticated structures that is time-consuming and demands high-ranking techniques for surface planarization and layer arrangement in multilevel stacking procedures seriously hamper the underlying applications. If the multi-spectral PIT resonances can be realized in one nanostructure, it will have wonderful benefits for applications in multi-wavelength biosensors. We can easily achieve multi-spectral PIT in one nanostructure by simply moving nanorods, which has narrow and high modulation depth in the resonances.

In this paper, we first investigate the optical properties of a hybrid nanostructure consisting of two parallel nanorods and a vertical nanorod (Δz = 0 nm). As the movement of the vertical nanorod increases, the spectrum changes from single PIT to double PIT (Δx ≠ 0 nm). The near-field coupling strength between nanorods is explained by the quasi-static interaction model. With moving nanorods, the quadrupole modes of the vertical nanorod are excited in transparent window. The coupling of pure dipoles and the coupling of a mixture of dipole and quadrupole modes lead to two transmission windows. Furthermore, by moving the bottom nanorod to increase the coupling strength (Δz ≠ 0 nm), the quadrupole mode of opposite phase of the intermediate nanorod is excited, resulting in one more mixture of dipole and quadrupole coupling modes to realize multi-spectral PIT. It is remarkable that the revealed theory clearly indicates that further transmission windows can be induced by altering the coupling strength between nanorods into our system.

2. Structure design

Figure 1(a) schematically shows the unit cell of metamaterial structure design, which consists of orientationally different silver nanorods (The surrounding media are assumed to be air for simplicity, ε_r = 1). The upper, middle vertical silver bar, lower and bottom silver bar are described as silver nanorod 1, 2, 3 and 4, respectively. The silver nanorod 3 and 4 coincide, which can be seen as nanorod 3 exists alone (Δz = 0 nm). The middle silver nanorod 2 is perpendicular to the upper nanorod 1 and lower nanorod 3. The lateral displacement of the middle silver nanorod with respect to the symmetry axis of the upper and lower silver nanorods is defined as Δx. Three sliver nanorods are placed differently than the previous one [9, 18-19], so that the exposure of the light source to the three nanorods is not the same. When the plane wave is vertically illuminated with x-polarization, the upper nanorod can be directly excited by light as a bright pattern. However, the middle vertical nanorod cannot be directly excited by light as the dark mode. Due to the presence of the upper nanorod, the lower nanorod cannot be directly excited by light, so it also is known as a dark pattern. To perform our numerical calculations by using the finite-difference time-domain (FDTD) method, the computational domain is truncated by perfectly matched layers (PMLs) in the z-direction and the periods in the x and y-directions are 500 nm.

 

Fig. 1 (a) Schematic of the proposed structure (the silver nanorods 3 and 4 coincide, denoted as Δz = 0 nm) (b) Δz ≠ 0 nm. (c) Schematic diagram of the plasmonic PIT structure with definitions of the geometrical parameters: The width of nanorods in the y direction is 40 nm, l₁ = l₂ = l₃ = l₄ = 140 nm, d = 20 nm, h₁ = h₃ = h₄ = 20 nm, h₂ = 40 nm. The upper, middle vertical silver bar, lower and bottom silver bar are described as silver nanorod 1, 2, 3 and 4, respectively.

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3. Results and discussions

3.1 Single-spectral plasmon induced transparency

We first present a single-spectral PIT in the three-dimensional structure of the metamaterial (Δz = 0 nm). In defect of structural asymmetry, that is, for Δx = 0 nm, there is only a single resonance perceptible in the transmission spectrum. The origin of this resonance can be due to the excitation of dipole-like plasmons inside the upper silver nanorod. This silver nanorod works as a dipole antenna and radiates forcefully to free space, causing a broad resonance in the spectrum. The coupling between nanorods starts once the structural asymmetry is introduced. As presented in Fig. 2(a), a micro transmission peak appears within the resonance profile. The modulation depth of the PIT transmission spectra peak can be tuned by increasing Δx, which corresponds to the amount of symmetry breaking. By increasing Δx, the transmission window grows in strength and becomes more prominent (Δx = 10 nm, 16 nm), and the blue shift and the red shift occur two dips (I and II), respectively. The above phenomena can be easily explained by the destructive interference between the two excitation pathways of the dipole modes of nanorods: the direct excitation by the incident plane light field and the excitation by the near-field coupling [5, 9]. The two pathways destructively interfere with each other, as a result of dramatically reducing extinction and intensifying transmission.

 

Fig. 2 (a)-(c) Simulated PIT transmission spectra for proposed structure (Δz = 0 nm) in dependence of lateral displacement Δx. I and II corresponds to two dips, respectively. (d)-(f) Field distributions E_y of the transmission peak and dips of the Δx = 16 nm transmission spectrum. The distribution E_y at the wavelength of 674 nm in (d), 704 nm in (e), and 735 nm in (f).

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In order to better understand the underlying physics, the electric field distributions (E_y) at the transmission dips and transmission peak are shown in Figs. 2(d)-2(f). For the two dips (I and II) in the electric field distributions E_y, dipole modes of the opposite phase are excited in the silver nanorods 1 and 3, forming an electrical quadrupole pattern, and the dipole mode of exactly opposite phase of silver nanorod 2 is excited in the two dips, so the two dips are produced by the interaction of the quadrupole and the dipole. For the electric field distribution E_y of transparent peak in Fig. 2(e), the in-phase dipole modes of silver nanorod 1 and 3 are excited in the electric field diagram (E_y), and the dipole resonance mode of middle silver nanorod is excited, so this transparent peak is a pure dipole resonance peak.

The PIT transmission window primarily depends on near-field coupling strength between nanorods, similar to the so-called Autler-Townes doublet in nuclear systems [22]. So as to intuitively show their coupling strength, we afford the interactive energy in the quasi-static approximation. As shown in Fig. (3), the interactive electric energy between the dipole moment of parallel nanorods and vertical nanorod can be depicted [23–25]. We can deduce that, when Δx increases (θ > 0), the interaction energy between the dipole moment increases (V_e > 0) [24]. Intuitively, the coupling strength increases between nanorods with the movement of the vertical middle nanorod, which enhances the interaction between the nanorods to strengthen the transmission.

 

Fig. 3 The Interaction of the dipole moment between parallel nanorods and vertical nanorod, where anti-parallel dipoles ${\overrightarrow{p}}_1$ and ${\overrightarrow{p}}_2$ represent the electric quadrupole moment and ${\overrightarrow{p}}_3$ denotes the electric dipole moment. The quasi-static interaction model is drawn based on the distribution E_y of dip II in PIT transmission spectra.

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3.2 Dual-spectral plasmon induced transparency

We further numerically and theoretically demonstrate that the resonance can also be changed by varying Δx. Interestingly, a weak transmission window II appears at the left dip (the original dip I) of the transmission spectral line when Δx = 22 nm, as shown in Fig. 4(a). As the asymmetry of the nanostructure is increased (Δx = 32 nm, Δx = 42 nm), the transmission peak II gradually increases shown in Figs. 4(b) and 4(c).With increasing Δx, the original dip I appears blue shift phenomenon in the spectral, as shown in Fig. 2. So the transparent peak II (the original dip I) meanwhile appears blue shift phenomenon in transmission spectral. The amplitude value of transparent window I is almost constant, and the original dip II continues to red shift resulting in the expansion of the transparent window I, with increasing Δx.

 

Fig. 4 (a)-(c) Simulated PIT transmission spectra in dependence of lateral displacement Δx (Δz = 0 nm). (d)-(e) Field distributions E_y of the two transmission peaks in the transmission spectrum (Δx = 42 nm).

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Similar to above, the electric field distributions (E_y) at the transmission peaks are shown in Figs. 4(d) and 4(e). From which, it can be seen that the two transparent windows generates with different coupling manners: for the transparent window I, dipole modes of all silver nanorods are excited, hence the transparent window can be understood as the result of interaction between the dipoles. For transparent window II, silver nanorod 1 as bright mode is coupled to dark mode silver nanorod 2 and 3. The electric quadrupole of silver nanorod 2 is excited at this time, and silver nanorod 3 is still dipole, so we can understand this transparent window II is made by the interaction between the electric quadrupole and the electric dipole. Comparing two transparent peaks, not only can we learn that a quadrupole mode of vertical nanorod 2 is excited but also we learn the dipole modes of upper and lower parallel silver nanorods appear completely opposite phase in two transparent peaks.

By further enlarging the lateral displacement Δx, the coupling strength between the dipole and quadrupole modes is successively increased. As a result, the transparency window II grows in strength and becomes more and more prominent as shown in Fig. (4). It also can be seen from the transmission spectral line that the two transparent windows have different linewidths, and the transparent window produced by interaction that is derived from a mixture of dipole and quadrupole resonance modes is narrower than the transparent window produced by the pure dipole resonance mode. Particularly, the narrow linewidths and high modulation depths of these PIT features indicate the drastically low loss in the quadrupole mode. With the increase of the lateral displacement Δx, the symmetry of the whole structure is seriously damaged, and the coupling strength between nanorods is enhanced, resulting in red shift of the high order mode (quadrupole mode) of the silver nanorod 2, so we observe the quadrupole mode of the nanorod 2 exists in transparent window II. Therefore, the increasing coupling strength plays an important role in the formation quadrupole pattern of nanorods 2 (dark mode), and nanorod 2 is essential for the formation of PIT window. The quasi-static interaction model successly explained the reason why the spectrum changes from single PIT to double PIT. It is noteworthy that the revealed mechanism clearly indicates that further transmission windows can be induced by increasing coupling strength between nanorods in our system.

3.3 Multi-spectral plasmon induced transparency

In order to show that the transmission window can be realized by increasing coupling strength in our system, we give the structure of multi-spectral PIT in Fig. 1(b). We move a silver nanorod 4 to achieve quadrupole excitation of the nanorod 2 to produce additional transparent windows, and other structural parameters are unchanged. The appearance of three transparent peaks is observed in the transmission spectrum, as shown in Fig. 5(a), and the electric field distributions (E_y) of the three transmission peaks are shown in Figs. 5(b)–5(d).

 

Fig. 5 (a) Simulated multi-spectral PIT (Δz = 36 nm, Δx = 50 nm). III, IV and V correspond to three transparent peaks (λ = 632 nm, 694 nm, and 753 nm). (b)-(d) Field distributions E_y of three transmission peaks in the transmission spectrum.

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We better understand the generation of three transparent peaks through the field distribution (E_y). It will be appreciated that the transparent window III is still formed by the interaction between the electric quadrupole and the electric dipole, with the same resonance phenomenon as the previous transparent window II. The difference of generation mechanism between the transparent window IV and the transparent window I is that the mode of silver nanorod 2 is changed from the dipole to the electrical quadrupole mode, so the coupling mode of transparent window is changed from pure dipole resonance to a mixture of dipole and quadrupole coupling resonance modes. It proves that silver nanorod 4 (dark mode) plays an important role in the formation of the quadrupole mode of silver nanorod 2 (dark mode). For the transparent window V, we can know the presence of an allowed magnetic dipole moment induced in the loops formed by the capacitance between the coupled lower nanorod 3 and bottom silver nanorod 4. The dipole modes of opposite phase are excited in the two nanorods, which corresponds to the excitation of the quadrupole mode.

When Δz = 0 nm, silver nanorod 2 supports a dipole mode and a quadrupole mode, then the spectrum corresponds to two transparent windows; when Δz ≠ 0 nm, silver nanorod 2 appears a dipole mode and two quadrupole modes, corresponding to three transparent windows. So the moving the silver nanorod 4 can make quadrupole of silver nanorod 2 is excited and produces more transparent windows. In a word, it is remarkable that the revealed mechanism obviously indicates that further transmission windows can be made by introducing excess quadrupole modes into our system. This promising future extremely highlights our three dimensional metamaterial concept for future PIT transparent windows applications operating at multiple frequency domains [26].

4. Conclusion

In conclusion, we have demonstrated multi-spectral PIT in three-dimensional metamaterials comprising of parallel nanorods and a vertical nanorod. The dipole and quadrupole modes of middle silver nanorod (dark mode) were excited with moving nanorods. The quasi-static interaction model revealed that the near-field coupling strength between nanorods increased with the movement of nanorod, which successfully explained the reason why the spectrum changes from single PIT to double PIT. In order to show that further transmission windows could be realized by increasing coupling strength, the bottom silver nanorod were moved to achieve quadrupole excitation of the middle vertical nanorod (dark mode) to produce additional transparent windows. It proved the multi-spectral PIT resonance could be realized by exciting additional quadrupole in our system. Furthermore, this work provides a way to obtain multi-spectral PIT, and will open a route for metamaterial applications such as sensing and slow light devices at multiple frequencies.