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Abstract
The surface plasmons that are excited by the multiple layer grating structures on the gold thin film are studied using the finite-difference time-domain method in this paper. The structure parameters’ effects on the coupling enhancement of surface plasmons are examined, and the structure design guidelines are given. It is found that the distance between the grating layers and the distance between the gratings and gold thin film are the key structure parameters for better cavity resonances. To have the stronger field enhancements of the excited surface plasmons for the multilayer grating structures, it is found that the width of the gratings should be smaller for the lower grating layers. The multiple layer gratings with proper structure designs can have better performances than single layer grating structure because the cavity effects can enhance the light coupling and more light can be coupled into the surface plasmons by more layers of grating. It is found that the maximum electric field intensity for five layer grating structures can be 163% of the case of the single layer grating structure in our simulations.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The surface plasmons have attracted a lot of attention in recent decades due to their electric field coupling, confinement, and enhancement for various applications, such as the plasmonic sensor [1,2], surface-enhanced Raman scattering (SERS) [3–5], plasmonic waveguides for controlling light in the scales much smaller than the light wavelength [6,7], anti-reflectance or light trapping for solar cells [8,9], and plasmonic nanolenses [10,11]. The surface plasmon polaritons (SPPs) are the types of the surface plasmons, and they are the collective oscillation of the electrons at the interface between the metal and dielectric materials [12]. However, they cannot be excited directly by the incident light because the phase-matching condition is not satisfied. The phase matching is an important issue to excite SPPs, and the waveguide coupling method has been proposed. For example, Dabos et al. demonstrated butt-coupled interface between Si3N4 waveguide and gold and it can be used in biochemical sensing in aqueous-surrounding medium [13]. Osowiecki et al. optimized the geometry parameter of the slot waveguide cavity, and the high sensitivity of 600 nm/RIU is shown in their simulation [14]. Briggs et al. reported the low insertion loss for polymer on gold to silicon on isolator waveguide with high coupling efficiency at telecommunication wavelengths [15]. Guo et al. showed the experimental results of high coupling efficiency to 80% between Ag and ZnO nanowires with high Q-factor [16]. Luo et al. studied how triangle metal taper mounted on dielectric waveguide can achieve vertical coupling [17]. Dabos et al. had developed the plasmon-photonic integrated circuits with CMOS plasmonics and photonics, which offset the disadvantage of high plasmonic propagation losses by the low-loss photonics in interconnect applications [18]. On the other hands, there are two general non-waveguide methods to excite the SPPs. One method is the prism coupling [19] and the other is the grating coupling [20]. The prism coupling method utilizes attenuated total internal reflection such as Kretschmann configuration and Otto configuration [21]. The light beam illuminates into a prism at a particular angle and the photon momentum on the metal surface should be satisfied the phase-matching condition at the interface between the metal and dielectric materials. The method of grating coupling is to give the additional momentum by the period grating or groove to the wave vector of incident light to satisfy the phase-matching condition. A remote grating structure was proposed to put the grating structures with some distance from the metallic thin film, and the SPPs can be excited on the metallic thin film [22]. The incident light was diffracted by the grating and the light with the diffraction angle which satisfies the phase-matching condition can be coupled into the SPPs. The features and benefits of SPPs excited by the remote grating structure is a narrow spectral bandwidth and the near-field intensity at the resonance wavelength can be higher than the conventional prism coupling [16].
The double layer grating was used to change the diffraction property and to improve the coupling efficiency [23–26]. Thus, the surface plasmons coupling and electric field enhancement in the multilayer grating structures are studied in this work. To achieve this, we consider the multiple layer grating structure to excite the SPPs and simulate their optical performances by the finite-difference time-domain method (FDTD). The effects of structure parameters on the electric field enhancement and bandwidth are also studied.
2. Simulation setup
For single layer grating structure, the incident light diffracted by the grating structure and coupled into the metallic thin film as the SPPs can be described by Eq. (1)
where m is the order of the diffraction grating mode, p represents as the grating period, λ0 is the coupling wavelength of the incident light, and ε1 and (λ0) are the dielectric constant of surrounding dielectric material and the real part of dielectric function of the metal, respectively. The SPPs can be excited when the phase-matching condition is satisfied. Figure 1 shows the simulation model of the multiple layer grating structure. The structure consists of n layers of periodic gold grating with a distance h from the gold thin film with a thickness l = 40 nm. According to Eq. (1), the coupling wavelength of SPPs depends on the period of the grating structures, the order of the diffraction grating mode, and the dielectric functions of metal and surrounding dielectric materials. Thus, to avoid the complexity, we fixed the same period p for all grating layers. Also, the thickness of all gratings are set as t in our design. The widths of gratings in different layers are denoted as w1, w2, and so on. The gratings are arranged staggered between the layers and the distance between two neighboring grating layers is d, which will be discussed in the next section. The Drude plus two-pole Lorentz model is used to define the dielectric constant of gold [27]. Because the semiconductor fabrication is matured, we choose SiO2 (silica) as the substrate with a dielectric constant εd in our design. The SiO2 has low imaginary permittivity in the wavelength we considered, which may reduce the loss of light in the substrate. The silicon substrate is also popular in the design of plasmonic applications. However, the silicon has higher loss in the wavelength range we considered. Thus, we choose SiO2 substrate in our simulations. The optical properties of SiO2 is chosen from the experimental data [28]. The permittivity above the gold thin film region is ε1, which is chosen as air in our simulations. The incident light source is plane wave which is TM wave and propagates normally from the bottom to the structure. The periodic boundary conditions are applied at the left-hand-side and right-hand-side boundaries, and the perfect matching layers are applied on the top and bottom boundaries. The SPPs are coupled by the diffracted light which comes from the bottom and is diffracted by the gratings and coupled into the SPPs on the top of the gold thin film when the phase-matching condition is satisfied. The simulations are performed by the Lumerical FDTD Solutions, which is a commercial software based on the finite-difference time-domain method.
Fig. 1 Schematic drawing of the multiple layer grating structure, where the distance between two layers of grating is d, the grating period is p, the grating widths in grating layer n is wn, the thickness of grating is t, the thickness of gold thin film above the SiO2 is l, εd is the permittivity of SiO2, ε1 represents the dielectric constant of air. The incident plane wave with x-polarization propagates in the positive y direction.
3. Double layer grating structure
In this section, we consider a double layer grating structure, which means there are only two grating layers in Fig. 1. According to the results of the single layer grating structure [22], the gold grating and the gold thin film can be treated as the cavity that results in the larger electric field intensity. When the double layer grating structure is applied, the transmission property can be controlled by tuning the distance between the two layers of grating [24]. Therefore, the study of effects of the distance between two layers of grating on the transmissions is essential to design the structure with stronger coupling of SPPs because there is a cavity effect between the two layers of grating [29]. To study the transmissions of double layer grating, we consider the structure which has only two layers of grating as shown in Fig. 2(a). Using the grating period p = 800 nm and the grating widths of first and second layer w1 = w2 = 200 nm, the transmission spectra are simulated for the different distances d between the two grating layers. According to Eq. (1), the resonant wavelength of the excited SPPs is about 815.5 nm in this case. The Raman spectroscopy is often interfered by the fluorescence scattering. In this wavelength range, the fluorescence scattering can be restrained. We change the distance between the two layers of grating d from 0 to 2000 nm, and the corresponding transmissions for different d are shown in Fig. 2(b). It is found that the transmission is periodic as d increases. The first four peaks are located at d = 347 nm, 608 nm, 893 nm, and 1198 nm, respectively. The double layer grating structure can be seen as Fabry–Pèrot-like cavity [29]. The equation is as the following:
where nd is the index of glass, λ0 is the incident wavelength, m = 0, 1, 2, … Here λ0 is 815.5 nm, and the refractive index of SiO2 is about 1.453 at the wavelength 815.5 nm. It means that λ0/2nd ≈ 280 nm, and it is similar to the difference in d between two adjacent transmission peaks.Fig. 2 (a) Schematic drawing of the double layer grating structure, where the distance between two layers of grating is d, the grating period is p, the grating widths of grating layers 1 and 2 are w1 and w2, respectively. The incident plane wave is x-polarization with wavelength 815.5 nm and propagates in the positive y direction. (b) Transmission for different d for double layer grating structure.
Figure 3 shows the magnitude of electric field distributions for the three peaks at d = 608 nm, d = 893 nm, d = 1198 nm in Fig. 2(b). We can find that the nodes of the standing wave increase as the distance between the two layer gratings increases. For the case of d = 893 nm, it has the strongest resonance between the two layer gratings. The stronger constructive interferences between two layers of grating is observed in Fig. 3(b). It means a better cavity-like effect between two layers for the case of d = 893 nm as comparing to the cases of d = 608 nm and d = 1198 nm. The two layers of grating can be considered as the two sides of cavity; however, it cannot be considered as a prefect model of cavity. As better resonances happen in the cavity, it can have stronger field enhancements. The whole physical mechanism of calculated transmissions in Fig. 2(b) is complicated, but this phenomenon of stronger interferences between two layers for the case of d = 893 nm can be corresponding to the reason of the high transmission at d = 893 nm in Fig. 2(b). For comparison, we also simulate the single layer grating and find its transmission is about 0.6 when the grating width is 200 nm. The energy transmissions between two layers of grating structure with d = 608 nm, d = 893 nm, and d = 1198 nm are higher than the transmission for the single layer grating. We choose the three largest transmission cases for double layer grating in this work to discuss further.
Fig. 3 Magnitude of electric field distribution, for (a) d = 608 nm, (b) d = 893 nm, and (c) d = 1198 nm at the resonance wavelength 815.5 nm.
The other important parameter of the double layer grating structure is the distance between the gold thin film and the upper grating layer, which is noted as h in Fig. 1. We simulate the double layer grating structures with different h from 0 to 1340 nm while the width of all grating is set as 200 nm and the grating period p is 800 nm. Figure 4 shows the electric field intensity enhancement, defined as |E|2, at the position about 5 nm above the gold film in our simulation for double layer grating structures. For comparison, the simulation results for the single layer grating structure with the same grating period and grating size is also shown in Fig. 4. The maximum value of single layer grating shown as black line is 159.8 at h = 1177 nm. The maximum value of the double layer grating structure with d = 893 nm shown as the green line in Fig. 4 is 231.5 at h = 892 nm. It can be seen that the electric field intensity enhancements for all peaks of the double layer grating structure with d = 893 nm are larger than the single layer grating structure. The peaks of the double layer grating structure with d = 893 nm are at h = 369 nm, 616 nm, 892 nm, and 1202 nm. It can be considered that the gold thin film and grating can form a Fabry–Pèrot-like cavity [14]. The incident light propagates into the system and diffracted by the grating. The first order diffracted light is coupled into the gold thin film and generates the SPPs. Some of the zeroth order diffracted light reflects back from the gold thin film or upper grating layers to the lower grating layers. The reflection of the zeroth order mode will be part of the origin zeroth order mode due to the light diffracting. It can be diffracted by the grating structure again to regenerate the first order mode diffracted light which can be coupled into the gold thin film as the SPPs. The first order mode of all diffraction can be seen as the geometric series with common ratio of reflectivity between two grating. The electric field is positive correlate to the quantity of regeneration. This is the reason why the double layer grating structure can have the stronger electric field intensity on the gold thin film because there are more SPPs coupled into the gold thin film.
Fig. 4 Electric field intensity enhancement of different grating structures, single layer (black line), double layer of d = 608 nm (red light), double layer of d = 893 nm (green line), and double layer of d = 1198 nm (blue line) with the width of grating as 200 nm and the grating period p as 800 nm.
The width of the gratings is also an important parameter in Fig. 1 because the grating size can affect not only the diffraction power but also the cavity effect between the gold thin film and the gold grating. It can be seen that the case of h = 1177 nm has the strongest field intensity for the single layer grating structure in Fig. 4. Thus, we fix the distance between gold thin films and grating at h = 1177 nm and increase the width of grating from 50 to 300 nm in our simulations. The wavelength of the incident light is still 815.5 nm. As shown in Fig. 5(a), the electric field intensities at 5 nm above the gold thin film for single layer grating starts from 7.5 for the grating width of 50 nm and reaches the maximum electric intensity of 174.6 for the grating width of 270 nm. In double layer grating structure, the widths of grating of the upper and lower grating are denoted as w1 and w2, respectively. We choose the structure d = 893 nm and h = 892 nm because of the maximum electric field intensity shown in Fig. 4. First, we fix w1 = 200 nm and change w2 from 50 nm to 300 nm and its corresponding electric field intensities at 5 nm above the gold thin film in double layer gratings are shown as the black line in Fig. 5(b). Then, to tune w1, w2 is fixed at the width of 200 nm for the maximum value of electric field intensity as shown in Fig. 5(b). In the following step, w1 is tuned from 50 nm to 300 nm. Its corresponding electric field intensities at 5 nm above the gold thin film in double layer gratings with different grating width w1 is shown as the blue line in Fig. 5(b), and it shows the electric field intensity of 243.0 at w1 = 211 nm. After tuning the structure parameters, the electric field intensity in single layer grating structure is 174.6 at h = 1177 nm, width of gratings = 270 nm, p = 800 nm, and the resonance wavelength is 815.5 nm. In double layer grating structure, the electric field intensity is 243.0 for the structure at h = 892 nm, w2 = 200 nm, w1 = 211 nm, p = 800 nm, d = 893 nm, and the resonance wavelength is also 815.5 nm.
Fig. 5 Electric field intensity in (a) single layer grating structure and (b) double layer grating structure as a function of grating width.
The relation between electric field intensity and gold thin film thickness is shown in Fig. 6. For the cases of thinner gold thin film, the SPPs coupling mode need to consider the SiO2 material and the coupling wavelength of SPP cannot be calculated by Eq. (1) but can be predicted in consideration of Insulator-Metal-Insulator structure. Because the wavelength of incident plane wave is fixed in Fig. 6, it results that the electric filed intensity is decreased as the gold thin film thickness is smaller because the coupling wavelength is shifted. On the other hand, for the thicker gold film, the electric field intensity is weaker due to the light cannot go through thick gold film for SPPs coupling. This phenomenon is also reported [16].
Fig. 6 Relation of electric field intensity and the gold thin film thickness for double layer structure.
4. Multilayer grating structure
From pervious section, it is found that the double layer grating can result in the stronger intensity of SPPs than single layer grating because of the higher transmission and the resonance between the two grating layers and gold thin film. Using the similar procedures as we discussed for the double layer grating structure in previous section, we add the third layer grating below the double layer grating structure and tune the width of all grating layers to achieve the highest electric field intensity on the gold thin film for the multilayer grating structures. The structure of multilayer grating structure is shown in Fig. 1. We add the grating layer by layer and optimize the width in every cases. The third layer grating is set below the second layer grating and the distance between the second and the third grating layer is chosen as 893 nm according to the results of the double layer grating structure in Fig. 3. The width of the third grating is denoted as w3. We first tune the width of the lowest grating layer to find the maximum value of electric field intensity at 5 nm above the gold thin film. After that, the width of the second and first lower grating layer is tuned. The fourth and fifth layer grating can be added by using the similar procedures of tuning structure parameters. After the tuning grating widths of every grating layer, the electric field intensity distributions of these cases are shown in Fig. 7. In single layer grating structure, the width of the grating layer w1 is 270 nm. In double layer grating structure, the widths in each layer w1 and w2 are 211 nm and 200 nm. In three layer grating structure, the widths in each layer w1, w2, and w3 are 211 nm, 190 nm, and 110 nm. In four layer grating structure, the widths in each layer w1, w2, w3, and w4 are 211 nm, 200 nm, 110 nm, and 50 nm. In five layer grating structure, the width in each layer w1, w2, w3, w4, and w5 are 211 nm, 198 nm, 120 nm, 51 nm, and 31 nm. The maximum electric field intensity for single layer to five layer grating structures are 174.6, 243.0, 273.5, 279.6, and 284.5, respectively. The electric field intensities excited by five layer grating structures are 163% of the electric intensity excited by the single layer grating structure. We can notice that when the additional grating layer is added, the width of the additional grating decreases gradually. It could be because the transmissions of incident light can be higher and more power of incident light can go into the system of the multilayer grating structure with smaller width of the lower grating layers. Also, more diffracted light can be coupled into the SPPs on the gold thin film. To prove this argument, we can find that the electric field intensity increases with little red shift as the grating layer increase as shown in Fig. 8(a). When the grating layer is more than four layers, the improvement of enhancing the electric intensity of SPPs is not significant. It is because the grating width of fourth grating layer is only 51 nm for four layer grating structure. The grating size is too small to have the good diffraction effect and the better resonance between the two grating layers, and it can also be observed for the multilayer grating structures in Fig. 7. We can see that there are stronger electric field resonances in the region between gold thin film and first grating layer and the region between the first and second grating layers. The electric field resonance is not strong between the fourth and fifth grating layers. The transmissions of multilayer grating structures with different grating layers are shown as the black line in Fig. 8(b). The electric field intensities at 5 nm above the gold thin film with different grating layer structures are shown as the blue line in Fig. 8(b). Generally speaking, the electric field intensity is stronger as the transmission is higher. To have the higher transmission in multiple layer grating structures, the width of grating in the lower grating layers must be smaller than that in the upper grating layers. The total loss of the structures in Fig. 7 can be calculated by the incident power divided by transmission and reflections. For the cases of Fig. 7, the corresponding losses normalized by the incident light power are 0.2249, 0.6335, 0.6104, 0.6190, and 0.6268, for single, double, three, four, and five layers of grating structures, respectively. It should be noticed here that more coupling to SPPs can reduce the transmission and reflection, which results higher loss in the calculations. The loss may also come from the electromagnetic field interacts with the grating structures. The designed multiple layer structures in Fig. 7 can be fabricated by using the similar fabrication processes proposed by Cheng et al. [30]. A single layer can be fabricated by below processes. The electron beam resist, like polymethyl methacrylate (PMMA), can be deposited on the glass substrate by spin coating method. The grating pattern can be designed and fabricated by the electron beam lithography. After etching the PMMA layer, the SiO2 can be placed on the spaces of etched PMMA.
Fig. 7 Electric field intensity distributions in (a) single layer grating structure, (b) double layer grating structure, (c) three layer grating structure, (d) four layer grating structure, and (e) five layer grating structure. The incident light is from the bottom to the top with the wavelength as 815.5 nm. The maximum intensities in different grating layer are 174.6, 243.0, 273.5, 279.6, and 284.5.
Fig. 8 (a) Electric field intensity at 5 nm above the gold thin film verse wavelength for the multilayer grating structures and (b) transmission and intensity of different layer grating structures, where the transmission is shown as the black line and the maximum electric field intensity is shown as the blue line.
The fraction of transmitted power of diffraction order modes for a grating can be calculated numerically by Lumerical FDTD Solutions, which is a commercial software based on finite-difference time-domain method. Because the first order mode can be coupled into SPPs in our designed multilayer structures, but zeroth order and higher order modes cannot couple into SPPs at the wavelength designed for the first order mode. The coupling wavelength can be calculated by Eq. (1). For zeroth order and higher order modes, the coupling wavelengths are different. Thus, the zeroth order and the higher order mode are not contributed to the SPPs couplings in our simulations. The results of the zeroth and first diffraction order modes for our grating structures are shown in Table 1. The higher order mode is not listed in Table 1, but the percentage of second order and higher order modes equations to one minus the percentages of zeroth and first order modes. It shows that zeroth order and first order modes are dominated modes in Table 1. The zeroth mode percentage decreases from lower layers to higher layer because the light was incident from bottom to top in multilayer systems. And the zeroth mode transmitted to lower layer can be excited to different order modes as it iterates with higher layers. In contrast, the first order mode percentage increases with the light propagated from lower layers to the high layers, it indicated the transformations between two modes. The first order mode percentages and the transmissions for three, four, and five layer grating structures are similar, which is consistent with the trend of the transmission.
Table 1. Summary of performance and structure parameters.
5. Coupling of SPPs by Gaussian mode source
In this section, we consider the Gaussian mode source, instead of plane wave light source. To be not too complicated, we take the structure parameters of the double layer grating of Fig. 7(b) for comparisons of different size of Gaussian mode sources. Because the common single fiber core size is about 8 μm, we consider the different full width at half maximum (FWHM) of Gaussian mode source as 2, 4, and 6 μm, respectively, in our simulations. The boundary conditions are set as the perfect matching layers along both x and y direction. The results are shown in Fig. 9. It is shown that with the lower FWHM, the electric field is more confined at the middle of the gold thin film, which may be regarded as the light source close to the remote grating structure. The propagation SPPs are not stronger but it can still be found on the top of gold thin film. The electric field intensity of SPPs on the gold thin film for the case of Gaussian mode sources in Fig. 9 is weaker than the case of plane wave sources in Fig. 7(b) because the light from the Gaussian light source is only interacted by fewer number of the grating and it results weaker SPPs coupling.
Fig. 9 Electric field intensity distributions under full width at half maximum of Gaussian mode source of (a) 2 μm, (b) 4 μm, and (c) 6 μm.
6. Summary
In this study, we use the FDTD simulation to investigate the excitation of surface plasmon polaritons by multiple layer grating structures. We optimize the distance between two grating layers and the distance between gold thin film and gratings. The simulation results show that the multiple layer grating structure can yield higher electric filed intensity than single layer grating structure. The reason is that the transmission of the incident light in multiple layer grating structure is higher than single layer grating structure. Another factor to affect the intensity of SPPs is the width of the grating. As the additional grating layer is added, the width of the additional grating decreases gradually for stronger electric field enhancement on the gold thin film. It is because the transmissions of incident light can be higher for the resonance between the grating layers and the gold thin film and more diffracted light can be coupled into the SPPs on the gold thin film for the system of the multilayer grating structure with smaller width of the lower grating layers. With careful design, it is found that the maximum electric field intensity excited by five layer grating structures can be 163% of the case of the single layer grating structure.
Funding
Ministry of Science and Technology, Taiwan (104-2221-E-002-079-MY3) and National Taiwan University (107L7829).
Acknowledgments
We are grateful to the National Center for High-Performance Computing, Taiwan, for providing us with the computation time and facilities.
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