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Abstract
A grating-coupled Otto configuration consisting of multilayer films including a few layers of graphene and a germanium prism is proposed. A sharp and sensitive Fano resonance appears when a graphene surface plasmon polaritons (GSPPs) mode from the graphene-dielectric interface couple with the planar waveguide (PWG) mode. We utilize the classical harmonic oscillator (CHO) to explain Fano resonance and study the influence of various parameters of the configuration on the reflection spectra. The highly sensitive sensor can be achieved by introducing detected materials into Otto structure. In addition, we investigated the effects from material loss arising in our designs. All of the simulations are performed by a finite element method (FEM).
© 2017 Optical Society of America
1. Introduction
Surface plasmon polaritons (SPPs), a kind of surface wave, propagate along the interface between the metallic and dielectric materials with exponentially decaying fields in both sides [1]. The plasmonic nanostructures, owing to their unique properties, are able to break through the diffraction limit and thus allow light to propagate in the sub-wavelength structure [2]. For instance, plasmonic has been mainly studied in metal-insulator-metal (MIM) waveguides [3, 4], photonic crystals [5, 6], metasurfaces [7, 8], etc., enabling the design of various micro-nano optical devices such as integrated photonic circuits [9], filters [10], sensors [11], etc. However, these devices are difficult to tune actively and have huge ohmic losses at visible wavelength, since they are usually made of noble metals. Graphene, a novel two-dimensional material which consists of a single layer of carbon atoms densely arranged in a honeycomb lattice [12, 13], has been widely applied as a novel plasmonic material in the infrared frequency regime [14]. Graphene exhibits excellent mechanical [15], electronic [16], and optical properties [17]. In particular, the carrier density and corresponding Fermi energy level of graphene can be actively modulated by external bias voltages or chemical doping, which lead to drastic variations in its optical properties [18]. In recent years, graphene structures have attracted great attention for Fano resonance and plasmonic waveguides [19, 20].
Fano resonances(FRs) [21] were first discovered in a quantum mechanical study of the autoionzation spectra of He atom, and they arise from the interaction of a narrow discrete resonance with a broad spectral line or continuum [22]. Fano resonances generate a sharp asymmetry of spectral absorption lines and an abrupt variation in amplitude and phase, when a broad resonance and a narrow resonance are coupled. Recently, a variety of Fano resonances have been explored [23–25]. Zheng et al. [26] have designed a modified multilayer thin film coupled Otto configuration to produce FRs in the mid-IR range, but it is hard to control due to its broad resonance caused by the SPhP mode. Guo et al. [27] proposed a tunable Fano resonance based on the interference between graphene surface plasmon polaritons (GSPPs) and a dielectric waveguide mode. However, the adjustment of sensitivity is still not achievable.
In this paper, we propose an improved grating-coupled Otto configuration consisting of multilayer thin films, including a few graphene layers and a germanium prism, followed by a numerical study of the dependence of its reflection spectra on geometrical parameters. The surface conductivity of the graphene layer can be dramatically tuned by the Fermi level EF. We utilize the finite element method (FEM) [28] to perform the simulation works. The results are simulated by using the MUMPS solver in COMSOL multiphysics. The waveguide mode shows a narrow resonance excited by the planar waveguide (PWG), while the graphene-dielectric interface excites a GSPP mode, bringing a broad resonance. The sharp Fano resonance appears when these two modes coupled. We use the classical harmonic oscillator (CHO) to explain Fano resonance and propose several modulation schemes to adjust the characteristics of the reflection spectra. Furthermore, we introduce a detecting layer in the Otto configuration and then achieve a sensitive sensor with high FOM, which can be used to design an environment-sensitive sensor.
2. Theoretical model and experimental scheme
As illustrated in Fig. 1(a), the grating-coupled Otto configuration presented in this paper is composed of multilayer thin films and a germanium prism, with a few layers of graphene inserted between the grating and detected materials. The structural materials and geometrical parameters are indicated in Fig. 1(a), and the thickness of each layer is assumed to be d1 = d2 = d3 = 2μm, and d4 = 0.5μm. In the following study, it was assumed that refractive indices np of the germanium prime, ns of the substrate and n2 of the PWG core were all set as 4. Refractive indices n1 of the cladding layer and n3 of the grating layer, made of CaF2, are both set to be 1.3. The detecting layer with refractive index nd was designed to place the materials waiting to be detected. The incident field in the mid-infrared regime was considered to be transverse magnetic (TM), exciting graphene and waveguide SPP modes.
Fig. 1 (a) 3D schematic illustration of grating-coupled and graphene-based Otto configuration. (b) The real part (black) and imaginary part (red) of relative permittivity of the monolayer graphene in relation to the Fermi level EF. (c)The effective refractive index NGSPP of different number of graphene layers in relation to the Fermi level EF. (d) Schematic illustration of the fabrication procedures of the grating-coupled and graphene-based Otto configuration.
According to the Kubo formula [29, 30], the optical properties of graphene monolayer are related to its complex surface conductivity σ, which is the sum of two sections: interband and intraband. These can be written as
where e, ћ and kB are elementary charge, reduced Planck's constant, and Boltzmann constant, respectively. In the graphene layer, the momentum relaxation time denoted by τ and the temperature of the environment denoted by T were set to 0.2 ps [31, 32] and 300 K, respectively.Moreover, the incident wavelength λ was set to be 10.6 μm throughout this work. The surface conductivity of graphene is controlled by its Fermi level EF, which can be dramatically modulated by the electrical gating. The EF is able to reach 1 eV in experiment by using top-gate electrical doping in extended graphene [33]. Figure 1(b) is based on Eq. (1) and Eq. (2) and the permittivity of the monolayer graphene, which can be described as εg = 1 + iσ/(ωε0dg). In the numerical simulation, the thickness dg of the monolayer graphene was set to 0.34nm.
One of the significant parameters of graphene is the GSPP effective refractive index NGSPP = kGSPP/k0, where kGSPP is the wave vector of SPP on graphene. The prism is able to excite SPPs only if its refractive index is larger than that of the GSPP. Realistically it is impossible to find a prism to overcome the huge momentum mismatch, as the monolayer graphene only can excite GSPP owning to its very large refractive index, especially in the infrared frequency regime. In this paper, in order to reduce NGSPP, a few layers of graphene were used to increase σ. It is reasonable to consider each graphene film as non-interacting monolayer when the number of its layers N is less than 6 (N<6). Thus a larger surface conductivity Nσ of the multi-layer graphene can be obtained [34]. For a single layer of graphene, the Fermi level EF can be expressed as ћvF(πns)1/2, where vF is Fermi velocity and ns is carrier density. In our work, we calculate the EF of a few layer of graphene as ћvF(πns/N)1/2 by assuming the charge is uniformly distributed among N layers [35]. This method has two advantages: it neither increases EF of the monolayer graphene, which may exceed the breakdown voltage, nor excites the GSPP in the longer wavelength range. Based on the discussion above, the dispersion relation for TM modes in the detecting layer-multi-layer graphene-grating system can be described as follows:
Here kGSPP can be calculated from the above equations. Furthermore, the NGSPP of different numbers of graphene layers shown in Fig. 1(c) (N = 4 and N = 5), which depends on the Fermi level EF, can be obtained. It is obvious that the effective refractive index of the PWG is a constant value when the thickness of the waveguide is unchanged. Comparing with N = 4 and N = 5 in Fig. 1(c), it is obviously that the NGSPP reduces significantly when the number of graphene layers increased, which enables the accurate matching of effective refractive index of the PWG. The GSPP mode can be controlled by changing the EF, and then it can be coupled with the PWG mode. Higher NGSPP makes few layer graphene excite SPP more difficult, because it needs higher EF when two modes coupled. Thus, the number of graphene layer was assumed to be N = 5 in the following discussions and in the numerical simulation, which is considered to reduce the momentum mismatch. From the above discussions, an active method for the emerging FR is proposed theoretically.
As shown in Fig. 1(d), we propose a potential fabrication procedure of the grating-coupled Otto configuration. First, the substrate was made of CaF2, spin coated with a polymer (methyl methacrylate) (PMMA) layer [36], and then the grating cylinders were formed by electron beam lithography (EBL) [37]. Following these stages, acetone was used to lift off the PMMA layer [38]. After obtained the grating, a chemical vapor deposition (CVD) method [39] is used to grow a monolayer graphene above the grating layer. Next, a few layers of graphene are grown on the monolayer graphene, by using a novel chemical vapor deposition (CVD) method [40] one by one. Then, a physical vapor deposition (PVD) method [41] can be used to form a 3-layer structure and this 3-layer structure was transferred to the grapehen-based grating. Besides, by using transferring technology, a germanium prism is placed under the detected materials. Finally, we transferred it to the ready-made multilayer structure, to obtain the grating-coupled Otto configuration.
There are many researchers using the classical harmonic oscillators (CHO) to illustrate the mechanism of Fano resonance in plasmonic nanostructures and metamaterial [[42,43]. In this paper, we use a simple model of two classical oscillators (CO) and propose an analogy between the CO systems. As shown in Fig. 2(a), there are two classical harmonic oscillators coupled by a weak spring. The oscillators 1 (OCL1) is driven by an oscillating external force which is similar to GSPP mode controlled by the incident light. The GSPP mode in system is related to OCL1, while the PWG mode to the oscillator 2 (OCL2). We can give the equation of motions of particles in the oscillators by using the same notation as those in [44] as follows:
where ɑ1eiωtis the external force acting on the OCL1, ν12 is a coupling constant, the x1 and x2 are displacement of particles, γ1 and γ2 are damping constants, and ω1 and ω2 are natural frequencies of the OCL1 and OCL2, respectively. In order to achieve steady-state solution, we assume x1 = c1eiωt and x2 = c2eiωt. Equation (9) and Eq. (10) can be obtained by some manipulations, as shown below:Fig. 2 (a) The schematic of the classical harmonic oscillators. (b) The simulated results of the PIT case and Fano case. (c) The theoretical results of the PIT case and Fano case.
The phase of the oscillators φ1(ω) and φ2(ω) can be defined through
The absorbed energy of OCL1 is given by
We take the time average of P(t) and then obtain the average absorption of per oscillation period
where Im[χ1(ω)] is the imaginary part of χ1(ω) defined through c1 = χ1(ω)ɑ1.When the GSPP mode is driven by the incident light in the prism with the incident angle θin, kx can be given by kx = (2π/λ)npsinθin, where np is the refractive index of prism. We simulate the plasmon-induced transparency (PIT) case and Fano case with specific parameters shown in Fig. 2(b) and Fig. 2(c), respectively. In the PIT case, the natural frequency of OCL1 is equal to that of OCL2 (ω1 = ω2 = ω0). While in the Fano case, kx value corresponds to θx as 58.04° and kx0 value corresponds to θx0 as 55.66°. Thus the natural frequencies are set as ω1 = ω0 and ω2 = 1.0275ω0. As shown in Fig. 2(d) and Fig. 2(e), we theoretically calculate the PIT case and Fano case. The coupling constant ν12 is assumed as 0.0055ω0, and the damping constants γ1 and γ2 are assumed as 0.03ω0 and 0, respectively. Compared with simulation results, we can well explain the Fano resonance by CHO system.
3. Results and discussions
Based on a zero-order permittivity of the effective media theory (EMT) [45], the effective refractive index of grating can be approximately expressed as
when a waveguide grating under TM illumination [46]. Here, f is the grating filling factor. nL and nH represent the low and high reflective index materials of the grating layer, respectively. In the following discussion, properly changing the period number and radio duty of grating is equivalent to tuning the grating filling factor f. Through such tuning, the effective refractive index of grating can be adjusted, thus controlling the GSPP mode.We changed the grating period numbers ranging from 4 to 9. As shown in Fig. 3(a), the FR spectra are noticeably and dramatically influenced by the period number of the grating. Interestingly, by varying the different grating periods, a dip and a peak of FR appear at 51.1° and 51.2°, respectively but with different reflection values. In order to quantitatively describe this influence, we defined a new parameter as k = (Rmax−Rmin)/0.1, where Rmax and Rmin represent the peak and dip values in the reflection spectra, respectively. Next, the slope k of FR was calculated with different period numbers, shown in Fig. 3(b), and it was found that the most obvious FR was achieved as the period number of the grating reached 7. For a better analysis, the period number of grating was set to be 7 in the following discussion.
Fig. 3 (a) Reflection spectra in relation to different period number of gratings. (b) The fitted line of the relation between the defined slope k and the grating period.
As discussed in the above theoretical section, it is known that the FR spectrum is originated from the coupling between the GSPP mode and the PWG mode. In this part, we discuss the influence of the Fermi level EF on these two modes. In Fig. 4(a), there is a notable angle shifting of the broad reflection dip with the rise of the Fermi level EF. When the GSPP mode is closer to the PWG mode staying at the same position, the coupling becomes stronger between them, which gives rise to an FR phenomenon. The coupling strength between the two modes is at the highest when EF increases to 0.75eV. To study the underlying principle further, the amplitude distribution of the normalized field distributions for GSPP mode and PWG mode is plotted and marked by “A”, “B”, “C”, “D”,and “E” in the second and third graphs of Fig. 4(a). As shown in Fig. 4(b), at the angle of 51.1°, the electromagnetic (EM) energy is mostly confined within the Ge planar waveguide and the CaF2 grating, indicating the PWG resonance mode. In Fig. 4(c), the EM energy mainly distributes around the graphene, which well demonstrates the resonated GSPP mode at the angle of 55.7°. When two such modes come closer to each other, a great FR spectrum can be seen in Fig. 4(a). The answer to the question of the origin of the FR is illustrated in Figs. 4 (d) and (e), where the two modes interfere with each other forming a FR line in the reflection spectrum. The amplitude distributions of the field |Ex| for “E” are the same as that for “C”, which represents the GSPP mode. Therefore, by applying a different EF, the FR effect can be adjusted in order to serve as an actively tunable sensor.
Fig. 4 (a) Reflection spectra in relation to different Fermi levels EF. (b)-(e) Distribution of the normalized electric field (|Ex|2/|E0|2) for the corresponding “A”, “B”, “C”, and “D” points are marked in Fig. 4(a).
As shown above, the PWG mode is nearly confined within the Ge waveguide, suggesting that the sharp dip in the reflection spectrum can be affected by the geometrical parameters of the waveguide. Therefore, we continued to study the influence of another factor, the waveguide thickness d2 on reflection spectrum. The thickness d2 was changed from 1.9μm to 2.4μm, with EF being 0.7eV and other parameters remained the same, as discussed above. The simulated results are displayed in Fig. 5, where the GSPP mode remains at around 55.7°, but the PWG mode moves to a larger angle as d2 increase. Just as analyzed before, the FR effect appears when the two modes come close to a point where they couple with each other intensively. Thus, there is another way to yield the FR type spectrum via adjusting the Ge waveguide thickness, though a passive one. Therefore, by varying EF or the Ge waveguide thickness, various FR lines can be achieved. Consequently, the analysis above actually offers a possible way to design a sensitive sensor.
And then, the effects of other geometrical parameters of the grating were investigated, as it was closely constructed to the Ge waveguide. For a better analysis, two new factors were introduced, named as ratio duty of height and width which were defined as fh = h2/h1 and fw = w2/w1, respectively, and the corresponding parameters are shown in Fig. 6(a). As can be seen in Fig. 6(b), the GSPP mode and PWG mode are affected slightly differently by varying fh and fw. The reflection dip of PWG mode is clearly deepened while the GSPP mode is slightly deepened as fh grows from 0.6 to 0.9. Besides, the GSPP mode shows a noticeable shifting in the process. By increasing fw, it presents a similar trend to that in Fig. 6(b). Based on these effects, GSPP and PWG modes with different strength can be accessed.
Fig. 6 (a) 3D schematic illustration of redesigned Otto configuration and the grating with its parameters. (b) The reflection spectra in relation to different ratio duties fh. (c) Reflection spectra in relation to different ratio duties fw.
After optimizing the parameter of sensor, we place the materials with the different reflective indices nd into detecting cavity to show the high sensitivity of our design. We estimate the sensitivity with intensity modulation as:
In order to compare the sensitivities of different types of sensors, the figure of merit (FOM) for the sensitivity can be describe as follow:
which is the sensitivity intensity in maximum value. As shown in Fig. 7(b), the Rmax can reach at 0.14 when the n is small as 0.001. Then we calculated that the Rmax/n can achieve to 140 RIU−1 in our design. The conventional Otto configurations [44, 47, 48] achieved sensor power up to 30-40 RIU−1 due to the limitation of the structure. Zheng et al. have proposed a graphene-based Otto configuration device as a sensing application with sensor power up to 120 RIU−1 [49], which is actually very high. In our work, we combined the graphene and grating in the Otto configuration and take the advantages to increase the sensing value to 140 RIU−1. The value of nd was set from 1.01 to 1.07 for different materials, and simulation results of the sensitive sensor are presented in Fig. 7(c).Fig. 7 (a) 3D schematic illustration of the improved Otto configuration with different detected materials. (b) Reflection spectra in relation to refractive indices from 1.05 to 1.052. (c) Reflection spectra in relation to different refractive indices nd of detected materials.
Finally, we discuss briefly the influence of material losses on the FR spectrum. In the discussion above, it was assumed that the waveguide structure is made of lossless materials; however, this is not possible to test in real experiments. Real waveguide layers fabricated in laboratory conditions are materials with loss, because of the physical conditions and the experimental methods [50]. Besides the intrinsic loss of materials, light scattering resulting from the rough surface is considered another kind of loss. The calculated results are shown in Fig. 8, where a better FR line can be seen with less waveguide material losses. Therefore, it is necessary to consider the effects from waveguide material and other kind of losses, when designing sensors based on our configuration.
4. Conclusion
In this paper, we presented a well-designed grating-coupled Otto structure consisting of multilayer thin films and a germanium prism, to achieve an FR line in the reflection spectra. Based on numerical analysis, it was found that the FR effect in the reflection spectra is due to the coupling between the PWG mode with a narrow resonance and the GSPP mode with a broad resonance. Moreover, we proposed several methods to tune the FR-type reflection spectra and numerically study their performance. By varying the Fermi level of graphene and the thickness of Ge waveguide, we were able to turn on or off the FR effect. Besides, we found a way to design a multifunctional sensor via adjusting the two kinds of ratio duties of the CaF2 grating structure. Finally, we introduced the detected materials into the structure to function as a sensitive sensor with high FOM. We hope that our proposed designs could pave new ways in the application of actively tunable environmental sensors.
Funding
National Natural Science Foundation of China (Grant Nos. 11504139, 11504140); Natural Science Foundation of Jiangsu Province (Grant Nos. BK20140167, BK20140128); Fundamental Research Funds for the Central Universities (Grant No.JUSRP115A15); China Postdoctoral Science Foundation (2017M611693); Key Laboratory Open Fund of Institute of Semiconductors of CAS (Grant No. KLSMS-1604); Training Programs of Innovation and Entrepreneurship for Undergraduates of Jiangnan University (Grant No. 2017376Y).
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