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Abstract
In this study, we investigate the impact of asymmetry on plasmonic-induced transparency (PIT) in structures with double dark modes. We have identified the ideal structural parameters for the single and double asymmetric dark mode structures, and it has been demonstrated that the usage of these factors improves sensing performance. The suggested structure is made up of a number of parallel black phosphorus nano-ribbons (BPNRs) with similar length, and it has been found that double asymmetric dark-dark systems creates a more stable and consistent transparency window, making it a strong contender to serve as a refractive index sensor. The proposed structure is simple and easy to manufacture, making it a more viable design for refractive index sensors for the terahertz range.
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1. Introduction
Destructive interference between two or more plasmonic modes of structure which results in formation of transparency window is known as plasmonic induced transparency (PIT), a phenomenon comparable to electromagnetic induced transparency (EIT). Typically, the incoming light excites one element of the structure directly, and as a result of coupling between the first element and the other elements the other components are indirectly excited [1]. This interaction the modifies optical response of the structure significantly, causing a transparency window to appear in the absorption spectrum. Several research fields such as: sensors [2], slow light [3,4], and optical filters [5,6] have made use of PIT phenomenon. PIT structures have primarily been created using metallic materials, however, there are some problems with many metal-based PIT systems such as fixed spectral response of the induced transparent window, high ohmic loss, and limited plasma lifetime [7]. As a result, new materials have been investigated to replace metallic materials due to their high loss and dimensions. In particular, two-dimensional materials like graphene have shown promising result [8–13].
Black phosphorus (BP) has recently attracted significant attention due to its exceptional optical and electrical properties. In contrast to graphene, BP material features a wrinkled hexagonal honeycomb structure, This unique structure of BP results in different effective mass in various directions, leading to anisotropic plasmonic responses [14,15]. The thickness of the material directly affects the BP band gap, with the bulk and monolayer having band gaps of approximately 0.3 eV and 2.0 eV, respectively. The distinctive physical properties of BP have prompted research on the effects of tunable plasmon-induced transparency using BP, which has gained attention in recent years [12,16,25,17–24].
Wu et al [17] utilized a single layer of BP to achieve a tunable PIT effect through destructive interference between bright and dark modes. In a similar vein, Han et al. [20] investigated a hybrid structure of graphene and BP to develop a PIT effect with outstanding tunability and a high Q-factor. The work of Xia et al. [11] is particularly insightful in this field, in their paper, they propose a novel approach to achieve all-order perfect impedance matching PIT by introducing a parameter called the phase difference. The phase difference captures the plasmon resonant phase details and indicates that PIT effects generally occur when the phase difference is approximately odd or even multiples of π. The authors demonstrate the effectiveness of their approach by constructing a model with two layers of stacked graphene nanoribbons (GNRs), which exhibits multi-order PIT effects. They find that the existence of these PIT effects is guaranteed by the significant freedom in choosing geometrical parameters. Furthermore, the high-order PIT effects demonstrate blueshifts, high sensitivities, and high Q-factors. Their study provides guidance for designing multi-structured PIT devices and paves the way for precise designs of on-chip devices capable of multifunctional applications. The results of the study offer a fundamental understanding of the existence of PIT and its potential applications in slowing light, sensing, perfect absorption, and controlling the state of light. However, the use of parallel black phosphorus nanoribbons (BPNRs) to generate highly controllable PIT effects has not received much attention. Thus, we aim to examine PIT effects based on dark-dark mode pairing.
In this paper, we investigate the mechanisms underlying the behavior of an asymmetrical BPNR structure to realize the PIT effect, with a particular focus on cross ribbon coupling and the formation of multiple Fabry-Perot cavities. The main driving factor behind the plasmon-induced transparency process is the coupling between the bright and dark modes of the structure, leading to destructive interference of surface plasmons and the creation of a transparency window in the absorption spectrum. Previous studies on such structures have primarily examined the effect of the distance between the modes on the location and size of the transparency window. However, by introducing structural asymmetry, we introduce a new degree of freedom that enhances the controllability of the transparency window. This enables the ribbons to interact with cross-interference, depending on the geometrical design, and strengthens or weakens the performance of the transparency window.
The structure mainly consists of four BPNRs. This micro-structure adopts the bright-dark-dark mode coupling method. When incident light is irradiated onto the structure, four BPNRs with similar material parameters are excited simultaneously. With an appropriate distance between the four ribbons and lateral offset, the coupling between BPNRs creates a transparency window. Investigating the influence of structural and environmental parameters on the transparency window is of interest and will be undertaken. While similar structures have been proposed for bright-bright and bright-dark configurations, to the best of our knowledge, few structures have been proposed for interaction between dark-dark mode configurations. Furthermore, most of the proposed structures of this kind are surface-based fabrications, making the presented layer-based structure comparatively simpler. The simple planar structure is widely used in various fields such as multi-channel filters, optical switches, modulators, and sensors [26–33].
The difficulty of fabricating multilayer black phosphorus structures with a relative ribbon shift must be acknowledged, as it poses a significant challenge in the fabrication process. The claim that our proposed design is easy to produce is based on a comparison with other surface-bound designs in the field of plasmon-induced transparency, which often require precise element design and can be limited in terms of dark mode interaction [31–34]. In contrast, the present design is a flat layered structure that is more conducive to dark-dark interaction and does not require as much precision in element design and on top of that fabricating layered structure is far simpler compared to complex surface bound structures. It is important to clarify that the proposed design is only can be considered simple to fabricate when compared to other multilayer BP designs with a relative ribbon shift that are surface bound.
In the next section, we will introduce a structure design with two dark modes and present the parameters of BP. Then, in the results and discussion section, we will utilize simulation results to find optimum values for the presented structure. We will examine the impact of changes in substrate refractive index on the single bright, first dark, and second dark modes. Subsequently, we will analyze single dark mode structures for both symmetric and asymmetric alignments. Finally, we will compare the performance of symmetric and asymmetric double dark mode structures in response to changes in substrate refractive index and incident light polarization.
2. Structural design and material parameters
To investigate the effects of interaction between two asymmetric dark modes, a simple structure has been proposed and shown in Fig. 1. The non-dispersive dielectric layer has a refractive index of 1.6 RIU. Parallel BPNRs are aligned along the X-axis, and their length in the Z-axis is unlimited. The overall size of the structure is set to w7 = 500 nm in the X-direction. Size of the bright mode ribbons are taken to be w1= w2 = 120 nm and dark mode ribbons are w3 + w4= w5 + w6 = 320 nm. The total height of the structure is h1 + h2 + h3 + h4 = 1440 nm. The BP thickness in simulation is set to 1 nm. At the bottom an Au layer with thickness of h4 = 50 nm has been implemented to act as a mirror.
Fig. 1. General schematic of the structure. Total width of the structure is equal to w7 = 500 nm. Total height of the structure is h1+ h2+ h3+ h4 = 1440 nm. Bright mode elements are taken to be of same size w1 = w2 = 120 nm. Separation between dark modes in X-direction is s2 = s3= 180 nm. Total width of the dark modes in each layer is w3 + w4 = w5 + w6 = 320 nm.
Numerical calculations are carried out using the Finite Difference Time Domain (FDTD) method. The boundary conditions have been set as periodic in the X-direction, while a perfectly matched layer (PML) is utilized in the Y-direction. In this design, the structure possesses six distinct BNPRs: two for the bright mode, two for the first dark mode, and two for the second dark mode. However, since the boundary condition of the structure is periodic in the X-direction, the ends of the ribbons on the left are connected to the ribbons on the right for the first and second dark modes. As a result, the presented design has four BNPRs in simulation: two for the bright mode, one for the first dark mode, and one for the second dark mode.
Since BP is an anisotropic material then permittivity components have a separate value in each direction [35]:
Sensitivity and FoM are calculated to quantify the sensing performance of the asymmetric double dark mode structure. Sensitivity and FoM are calculated by: $S = \frac{{\Delta \lambda }}{{\Delta n}},\textrm{ }FoM = \frac{S}{{FWHM}}$. Here $\Delta \lambda $ is the shift in position of the absorption peak, $\Delta n$ is change in the refractive index that causes that shift and FWHM is full width at half maximum of the peak.
3. Results and discussion
Examining the impact of the separation between the two ribbons that make the bright mode on the absorption peak is important as the first step. Figure 2 (a) illustrates the result of bright mode ribbons absorption when they are initially positioned close together before being shifted toward the simulation space's edges. It is observed that when the distance between two ribbons is higher than s1 = 25 nm, there is no discernible impact on the structure's ability to absorb light, but for values less than s1 = 25 nm, the width of the absorption band narrows and the absorption peak moves toward shorter wavelengths. When the same simulation is conducted with presence of double asymmetric dark modes, the results of which are shown in Fig. 2 (b), It is observed that the transparency window shrinks as the distance between bright modes increases. Moreover, the absorption peak in the 6000 range gets stronger as separation increases. As a result of these observations, the value of separation between bright modes is set at s1 = 60 nm.
Fig. 2. Effect of ribbon separation in X-direction (s1) on absorption peak of bright mode on: a) lone bright mode and b) bright mode with double dark modes. It can be observed from presented results that in absence of dark modes, separation between bright mode elements in X-direction result in a steady absorption peak in 7250 nm wavelength with addition of dark modes result in formation of a new absorption peak in 6000 nm and shift the pre-existing absorption peak to higher wavelengths.
Impact of separation between bright and dark modes in the Y-direction is depicted in Fig. 3. It is clear that the absorption peak is extremely sharp for separations lower than approximately h1 = 7 nm, but as the separation increases, it is observed that a transparency window is formed in range of h1 = 10-60 nm. For the current simulation, a separation of h1 = 40 nm between the bright and the first dark mode was chosen. Taking into consideration Fig. 2(a), it is evident that the absorption performance of the bright mode ribbons only affects each other within the range of 0 to 25 nm. Beyond this range, there is a consistent absorption behavior of the bright mode ribbons at a wavelength of 7000 nm, indicating the absence of pairing between the phosphorene ribbons at distances greater than 25 nm. However, an interesting observation can be made from Fig. 2(b), which explores the effect of the movement of bright mode ribbons in the X-direction in the presence of dark modes. It is apparent that dark mode ribbons located at distances greater than 25 nm from the bright mode ribbons still influence the absorption performance of the structure. This suggests that the 25 nm is not the coupling distance range limit for phosphorene. Figure 3(a) further demonstrates the impact of separation in the Y-direction on the absorption behavior of the structure, specifically the separation between the first dark mode and the bright mode. This provides evidence that the coupling range is directly related to the width of the ribbon. As the ribbon width increases, the coupling range of the structure also increases. Considering these points, it can be concluded that the pairing distance for the ribbons in phosphorene has a direct relationship with their width which can be explained by taking into consideration that by increasing the width of these mono-layer ribbons, carrier density of the ribbon is also increased. As a result, since the bright mode ribbons, have a smaller width, they exhibit a lower coupling range of approximately 25 nm compared to longer BP ribbons.
Fig. 3. effect of Y separation between a) bright and first dark mode (h1), with increase of separation between dark and bright mode coupling between them weaken and two absorption peaks slowly merge together, and b) first dark mode and second dark mode, with increase of separation between two dark modes we first observe formation of multiply transparency windows in range of 10-50 nm then in range of 50-75 nm these transparency windows merge together and for separation greater than 75 nm structure only possesses a single transparency window.
Another point of interest is that the current structure consists of three cavities. The first cavity is formed between the second dark mode and the gold mirror, with a height of h3. The second cavity is created between the first dark mode and the gold mirror, with a height of h2 + h3. The third cavity exists between the first and second dark modes, with a height of h2. Figures 3(a) and 3(b) illustrate the changes in absorption performance for different distances in the Y-direction relative to the bright mode for first and second dark modes respectively. These diagrams provide insight into the effect of cavity height on the absorption of the structure. The total height of the structure is 1440 nm, and the height of the gold mirror is 50 nm, resulting in a distance of 1390 nm between the bright mode and the gold mirror. In Fig. 3(a), we observe the behavior of the structure as the size of the cavity between the first dark mode and the gold mirror changes from 1390 to 1290 nm. We note that within the range of 1385 to 1330 nm, a transparency window is formed, and the structure reaches its optimal state at a size of 1360 nm. For sizes between 1330 to 1390 nm, the behavior of the structure is similar to the case when the dark mode is not applied. Figure 3(b) illustrates the behavior of the cavity between the first and second dark modes for range of 0 to 200 nm, and the cavity between the second mode and the gold mirror for heights ranging from 1350 to 1150 nm. By increasing the size of the cavity between the two dark modes from 0 to 60 nm and decreasing the size of the cavity between the second row of the dark mode and the gold mirror from 1350 to 1290 nm, several transparency windows appear in the absorption spectrum of the structure. However, by increasing the size of the cavity between the dark modes and reducing the cavity between the second dark mode and the gold mirror, a larger and more stable transparency window is created. From this point it can be said that, formation of multiple Fabry-Perot cavities with different heights inside the structure is an important mechanism that affects the location of the transparency window. By accurately determining the distance between each row of the dark modes and the gold mirror, it becomes possible to enhance more than one frequency within the window, resulting in multiple transparency windows in the structure. The arrangement of the dark modes can also be made asymmetric, covering the entire space below the bright modes, which helps to reduce power loss and improve the overall performance of the structure and on top of that add a new degree of freedom to the structure design.
In next step we are interested in finding the optimal position of dark modes in X-direction. In the beginning we set first dark mode fully in the left side of simulation space and second dark mode in the right side (w4 = w5 = 0 nm and w3 = w6 = 320 nm and s2 = s3 = 180 nm). Afterward we shift each of the dark modes in opposite directions until they are fully on the other side (w4 = w5 = 320 nm and w3 = w6 = 0 nm and s2 = s3 = 180 nm). Effect of this motion on the absorption performance of the structure is observed in Fig. 4. It can be inferred from presented results that performance of structure remains consistent regardless of which dark mode is placed on the right or left side of the structure. This can be seen as a result of coupling between two dark modes negating the effect of their location. It can also be seen that as dark modes move toward a more symmetric position (w4 = w5 = 160 nm and w3 = w6 = 160 nm and s2 = s3 = 180 nm) the main transparency window is closed and in its place a less optimal window is formed. Of note is the fact that main absorption peak shifts toward lower wavelengths. In order to achieve optimum performance for present structure values of w3, w4, w5, w6, s2 and, s3 have been set to w4 = w5 = 37 nm and w3 = w6 = 283 nm and s2 = s3 = 180 nm. As the simulation boundary behaves periodically in the X-direction, moving the dark modes in that direction effectively simulates interaction between four dark modes with the bright mode and each other. It is observed that due to the asymmetry the structure's transparency window's placement shifts, which can be used to design and create a more dynamic transparency window.
Fig. 4. a) Effect of shifting dark modes relative to each other in X-directions, with decrease in the size of w3 and w6, the size of w4 and w5 increases and as result, the separation between dark mode elements in X-direction are maintained this shift in X-direction give the structure another degree of freedom. b) Absorption performance of the structure for X shifts equal to 53.5, 106.5 and 160 nm.
The performance of bright mode and each of the dark modes separately are examined in the next step. Figure 5(a) shows that bright, first dark and, second dark modes individually exhibit an absorption peak in the 7000-8000 nm region, with the bright mode's peak being significantly stronger than the peaks of the two dark modes. However, when first and second dark modes are placed in the structure simultaneously in absence of bright mode, absorption peak shifts from 7000-8000 nm to 8000-9000 nm range. It can be deduced from Fig. 5(a) that the position of dark modes in Y-direction have minor effect on their absorption performance. Next, the performance of the structure for the case of one bright mode and one dark mode in absence of the other dark mode is evaluated. A comparison of the structure's absorption capabilities for the first and second dark modes is shown in Fig. 5(b). As demonstrated, the first dark mode is able to form a transparency window in the 6000-7000 nm range. However, the second dark mode, which is located farther from the bright mode is far more weakly coupled with the bright mode, and hence is unable to form a transparency window. Furthermore, it is observed that due to stronger coupling between bright mode and first dark mode, absorption peak of the structure shifts toward lower wavelengths. Finally, it is observed in Fig. 5(c) that when both dark modes are used simultaneously as the result of the interaction between the two dark modes, transparency window become wider and minimum absorption of the transparency window decreases.
Fig. 5. Absorption diagram for a) comparison between bright and dark modes, dark modes can absorb the incident light but is weak compared to absorption of bright mode and as such can be neglected. b) the asymmetric dark single mode for the first and second dark mode, first dark mode stronger coupling with the bright mode result in formation of a transparency window and shift of the absorption peak to lower wavelength but coupling between second dark mode and bright mode only results in increase of absorption of structure in 6000-6500 nm wavelength range. c) the asymmetric double dark modes, with implementation of both dark mode transparency window shifts slightly to higher wavelengths and width of the window is increased.
Figure 6(a) shows that, as a result of change in refractive index of substrate, absorption peak of the bright mode is expanded and shifts toward lower wavelength. It can also be deduced from Figs. 6 (b) and (c) that both dark modes react in almost the same manner with regard to changing of refractive index of substrate. But, in case of double asymmetric dark modes, Fig. 6(d), it is seen that distinctive absorption peak that is present for both lone dark modes in 1- 1.25 RIU range is missing, which can only be result of destructive coupling between these two modes. This shows that if dark modes are placed together they effectively disable each other’s absorption in 1-1.25 RIU range, even before placement of bright mode. It's important to note that changes as the refractive index of the substrate increases the absorption peak wavelength location. For dark modes, the peak initially forms near 12,000 nm and shifts towards lower wavelengths. Unlike the bright mode, the shift for dark modes doesn’t possess a linear shape and instead, it has an exponential shape. Additionally, when dark modes are coupled together, the peak's performance improves compared to the results of a single dark mode.
Fig. 6. Absorption peak shift for a) bright mode, b) first dark mode, c) second dark mode and, d) double dark mode without bright mode as result of change in substrate refractive index. It should be pointed out that as refractive index of substrate increases absorption performance of second dark mode become superior to the first dark mode as observed by comparing results of (b) and (c).
Figure 7 shows the effect of change in refractive index of substrate on the position of absorption peak when one dark mode interacts with bright mode. Using these figures, it is possible to compare structures containing a single symmetric dark mode, a single asymmetric first dark mode, and an asymmetric second dark mode in terms of their absorption peak shifts in Fig. 7(a), Fig. 7(b), and Fig. 7(c) respectively. As can be observed in Fig. 7(a), up to 1.5 RIU, the symmetric structure has two transparency windows, at 7500-9500 nm and 9500-12000nm. However, at higher refractive indices, the performance of the symmetric structure shifts to a higher wavelength and the transparency window becomes less effective. From result of the first asymmetric dark mode, Fig. 7(b), it can be discerned that it is not possible to create two transparency windows for refraction index greater than 1.25 RIU, and the existing window is extremely narrow and does not cover the full range that the symmetrical dark mode covers. Interestingly Second dark mode only succeeds in producing a transparency window in the refractive index range of 1-1.5 RIU. However, as refractive index increases, it can be seen that second absorption peak broadens and first peak amplitude weakens, and effectively nullify the transparency window.
Fig. 7. Absorption peak shift for a) bright mode and single symmetric dark mode, b) bright mode and single asymmetric first dark mode, c) bright mode and single asymmetric second dark mode as result of change in substrate refractive index. Symmetric system creates two transparency windows for refractive index between 1-1.5 RIU and for index higher than 1.5 is comparatively weak. First asymmetric dark mode creates a consistent window between refractive index of 1-2 RIU and only a single transparency window is form as result of the second asymmetric dark mode in range of 1-1.5 RIU.
Now we must compare the results of the symmetric double dark mode with asymmetric double dark mode structures. From Fig. 8(a), it is observed that double symmetric structure is not able to effectively form any transparency window for refractive index greater than 1.25 RIU. Which means that using two symmetric dark modes of the same size result in inferior performance compared to single symmetric dark mode. On the other hand, Fig. 8(b) shows that the double asymmetric dark mode structure, has a transparency window that is active in the range of 1-2.25 RIU, The only structure that can provide comparable results is the single symmetric structure. However, due to the change in wavelength range of transparency window for refractive index above 1.5 RIU and weaker performance of second transparency window for refractive index greater than 1.5 RIU, the double asymmetric structure is more effective for practical applications.
Fig. 8. Absorption peak shift for a) bright mode and double symmetric dark mode as result of change in substrate refractive index, as result of coupling between symmetric dark modes absorption peaks on either side of main absorption peak shift toward the central peak and their efficiency increases. b) bright mode and double asymmetric dark mode as result of change in substrate refractive index, because of the coupling between asymmetric dark modes, absorption between two main dark modes decreases and as such forms a better transparency window.
The effect of symmetric and asymmetric arrangements on the structure's performance under different polarizations is the last parameter that has been investigated. It has been reported that a structure with BP bright mode and symmetric single dark mode exhibits a transparency window in the 7000-9000 nm range with a maximum width of 2000nm as result of change in polarization [19]. It has also been reported that for symmetric double dark modes, where each dark mode possesses a different sizes, change in polarization results in a transparency window in the 8000-13500 nm range with a maximum width of 5000 nm [22]. The results of a similar test have been presented in Fig. 9 for the symmetric structure (a) and the asymmetric structure (b) with the same-sized double dark mode structure. It is observed from presented results that the transparency window for the symmetric structure is formed in the range of 5500-6500 nm with a maximum width of 1000 nm. On the other hand, for the asymmetric structure, the transparency window lies in the range of 6000-7500 nm with a maximum width of 1500 nm. It should also be noted that the transparency window that is formed by symmetric structure is far less consistent for different polarizations and as polarization angle increases the transparency window expand to match result of asymmetric structure but even then, transparency window of the asymmetric structure is superior to transparency performance of symmetric structure.
Fig. 9. a) Diagram for the symmetric double dark mode absorption performance in relation with polarization, b) Diagram for the asymmetric double dark mode absorption performance in relation with polarization. While main peak for symmetric system are farther apart but the absorption intensity between them is fairly stable and consistent but for asymmetric system a clear dip between two main absorption peaks can be noticed.
Sensitivity of first peak and second peak and transparency valley have been presented in Fig. 10(a) and FoM performance of the first and second peak have been shown in Fig. 10(b) and these values have been compared to similar PIT Sensors in Table 1.
Fig. 10. a) Sensitivity performance of first peak, second peak and valley for refractive index from 1 to 3 RIU. With increase of the refractive index sensitivity of peaks and valley show an almost linear shift to lower values. b) FoM performance of the first and second peaks. As can be seen while Second peak’s FoM decreases consistently first peak’s FoM is remains around 40 1/RIU.
Table 1. Comparison with similar published works
The main advantage of the presented structure is, coupling between first dark mode with second dark mode and excitation of second dark mode through the first dark mode. First dark mode is excited by the bright mode but since second dark mode isn’t directly excited by the bright mode it can be used to enhance or dampen the plasmonic response of the first dark mode by tuning the separation between first and second dark modes, which results in formation of a tunable sensor.
4. Conclusion
The development of advanced structural parameters for single and double asymmetric dark mode structures represents a significant breakthrough in the field of sensing technology. Through meticulous calculations of sensitivity and FOM, the superiority of the asymmetrical structures in enhancing sensing performance has been unequivocally demonstrated. Further studies on the transparency window of both symmetric and asymmetric structures, as well as light polarization, have revealed that the proposed asymmetric structures offer unparalleled advantages over equivalent symmetric structures. Remarkably, the double asymmetric dark-dark systems have been found to possess a remarkably stable and consistent transparency window. This unique property enables the presented structures to achieve much higher sensitivity performance compared to similar designs, making them an exceptionally strong candidate to serve as a refractive index sensor. As a result, it is now a more viable and practical option for refractive index sensors within the terahertz range. Our investigation also highlights the importance of cross ribbon coupling and the formation of multiple Fabry-Perot cavities in the behavior of the structure. By designing the structure asymmetrically, we have added another degree of freedom to our design and now it is possible to increase the controllability of the transparency window and improve the overall performance of the structure. These findings could have potential applications in the development of novel photonic devices. These breakthroughs in sensing technology offer an exciting new avenue for future research, with the potential to revolutionize a broad range of applications in the field.
Funding
Deputy for Research and Technology of Shahid Beheshti University (600/74).
Disclosures
The authors declare no conflict of interest.
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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