A novel dielectric-loaded surface plasmon polariton (DLSPP) waveguide with an air nanohole within a high-index dielectric ridge is proposed and analyzed. It is demonstrated by simulations that the introduced air nanohole could strongly modify the modal behavior, and it could alleviate the transmission loss caused by the high-index ridge with rather small sacrifice in the mode area. Under certain geometric parameter ranges, a shallow and wide air nanohole at the metal surface could result in strong local field enhancement while improves the figure of merit (FOM). The proposed structure could enable the realization of the DLSPP waveguide with a high-index ridge to achieve subwavelength mode confinement with relatively low transmission loss.
©2010 Optical Society of America
Guiding and confining light at the subwavelength scale has become an ever-increasing demand for future high-density photonic integrated circuits . Waveguides and devices based on the surface plasmon polariton (SPP) have been proposed and demonstrated by leveraging the propagation of SPP at the metal/dielectric surface as a novel guiding mechanism . In the past studies of SPP devices, many efforts have been made to balance the tradeoff between the mode confinement and the inevitable propagation loss due to the Ohmic loss in the metal . Recently, dielectric-loaded SPP (DLSPP) waveguides employing a dielectric ridge directly on a metallic surface have been proposed and demonstrated . Compared with those previously proposed long-range SPP waveguides , metal slot waveguides  and channel SPP waveguides , DLSPP waveguides could effectively provide relatively tight confinement of light while maintaining relatively long propagation distance . Many DLSPP waveguide-based devices such as the directional couplers , splitters  and resonators  have also been demonstrated.
Conventional DLSPP waveguides utilize low-index materials, such as polymethylmethacrylate (i.e. PMMA, n~1.5), as the material of the ridge. It enables the relatively low-loss transmission but sets a limit on the downscaling of the waveguide structure. Typically, in order to ensure moderate propagation distance under the single-mode condition, the width and height of the dielectric ridge are as large as ~600 nm for waveguides at 1.55 μm wavelength . This, along with the near-micron-large-mode-size , presents challenges for true subwavelength integrations. Introducing a high-index dielectric ridge could enhance the field confinement , but the associated increase in the transmission loss is often unacceptable [13, 14]. One widely studied method for loss reduction in plasmonic waveguides is employing a low-index buffer layer between the metal and high-index region [14–17], and the associated hybrid plasmonic mode could enable low-loss transmission with sub-wavelength field confinement. Similarly, introducing air gaps in the vicinity of metals have been found to be useful for lowering the loss in the long-range SPP waveguides . On the other hand, substructures consisting of air in the core have been proven to enhance the field confinement in photonic crystal fibers  and dielectric slot waveguides .
In this paper, we propose a novel DLSPP structure consisting of a high-index dielectric ridge that has an air nanohole at the interface of the metal and the ridge. The size of the ridge can be greatly reduced due to its higher index. The air nanohole on the other hand helps to reduce the transmission loss. Further field enhancement could also be achieved in the nanohole [15, 17, 21]. Under optimized parameters, a subwavelength mode area could be realized while the propagation loss is also reduced. This relatively simple structure could be compatible with the conventional lithography fabrication process. Similar to the fabrication process of the well-studied air-filled slot waveguides , electron-beam lithography and etching, followed by the chemical-vapor deposition process of silicon, could be used.
2. Geometry and modal properties of the proposed DLSPP waveguides
For comparison with the proposed structure, a DLSPP waveguide consisting of a high-index dielectric ridge with a width of w and height of h sitting directly on top of a semi-infinite metallic substrate is shown in Fig. 1(a) . The materials considered are silver (Ag) for the substrate, silicon (Si) for the ridge, and silica (SiO2) for the cladding, respectively. Our proposed DLSPP waveguide is shown in Fig. 1(b), where a nanohole filled with air is incorporated at the bottom center of the high-index ridge. Its width and height are wa and ha. In our simulations, the wavelength is set at λ = 1550 nm. The permittivities of SiO2, Si and Ag are εc = 2.25, εd = 12.25 and εm = −129 + 3.3i , respectively. The width and height of the dielectric ridge are w = h = 200nm, and they are chosen to ensure that the corresponding conventional DLSPP waveguide supports a single TM SPP mode . The modal properties are investigated by means of the finite-element method (FEM) using COMSOLTM. The eigenmode solver is used with the scattering boundary condition. Convergence tests are done to ensure that the numerical boundaries and meshing do not interfere with the solutions.
Our results indicate that the proposed DLSPP waveguide can support a fundamental quasi-TM plasmonic mode under a wide range of geometric parameters of the nanohole. As an example, Ey distributions and the cross-sectional curves of Ey of the fundamental plasmonic mode of the conventional and proposed DLSPP waveguides with wa = 100nm and ha = 30nm are shown in Fig. 2 and Fig. 3 . For comparison, the field distribution of a pure dielectric structure with a SiO2 substrate is also drawn in Fig. 2(c). As shown in Fig. 2(a), the 200nm × 200nm Si ridge is capable of effectively confining the field due to its high index. By introducing the air nanohole in the ridge, the tails of the field distribution spread out slightly more, as the low-index nanohole slightly weakens the field confinement capability of the ridge. However, in either case, subwavelength mode confinement can be achieved, because of the smaller, high-index ridge. The effective mode sizes (Deff), defined as the diameter of the effective mode area (Aeff), of the structures in Fig. 2 (a), (b) and (c) are 0.24, 0.28 and 1.56 μm, respectively. Aeff is calculated using , where, to accurately account for the energy in the metal region, the electromagnetic energy density W(r) is defined as [24, 25]:21]. Despite the increase of field intensity in the nanohole it is noted that the field distributed in the metal is significantly reduced as seen in Fig. 3(e), and the propagation length Lp defined as λ/[4πIm(Neff)] is nearly doubled (from 17 to 31 μm), where Neff is the modal effective index.
The dimensions of the nanohole strongly influence the modal properties. Figure 4 shows the electric field distributions at different nanohole heights with wa fixed at 100nm. It is noted that smaller ha would cause stronger field enhancement for both horizontal and vertical directions in the nanohole region. At larger ha’s, the field enhancement is less obvious, and more electric field is distributed in the upper part of the high-index ridge, leading to an increased mode area. Deff of the structures in Fig. 4 (a)-(d) are 0.27, 0.28, 0.29 and 0.33 μm, respectively. Comparisons of Ey distributions at various nanohole widths in Fig. 5 illustrate a similar trend but the resultant changes are less obvious. Widening the nanohole leads to a slight increase in the spreading of the field distribution. Deff of the structures in Fig. 5 (a)-(d) are 0.25, 0.27, 0.28 and 0.33 μm, respectively.
To further quantify the field confinement and enhancement in the air nanohole region, the normalized optical power (NOP) and normalized average optical intensity (NAOI) in the nanohole are compared at different widths and heights, as shown in Fig. 6 . The NOP is defined as the ratio of the power inside the nanohole to the total power of the waveguide. It increases at bigger ha when wa is small. However, when wa is relatively large (e.g. 100nm or 150nm), the NOP may decrease at bigger ha, as a large air hole significantly weakens the field confinement capability of the ridge. The NAOI is given by NOP divided by the nanohole area and normalized to the peak intensity in the dielectric ridge of the corresponding traditional DLSPP waveguide shown in Fig. 2(a). It is shown that when ha is small (e.g. <10nm), the calculated NAOI exceeds far beyond 1, especially when the nanohole is wider, indicating a several-fold enhancement in the optical intensity over that in the traditional DLSPP. However, as ha is getting larger (e.g. >30nm), the NAOI at larger wa drops more rapidly, due to the greater weakening of mode confinement in the nanohole region.
Neff, Lp, the normalized mode area (Aeff /A0) and figure of merit (FOM) of the SPP mode of our proposed structure with different nanohole sizes are shown in Fig. 7(a)-(c) . A0 is the diffraction-limited mode area and defined as λ2/4. FOM is defined as the ratio of Lp to Deff . Figure 7(a) shows that both of the modal effective index and propagation loss decrease monotonically when the nanohole size (width or height) increases. As shown in Fig. 7(b), at relatively smaller nanohole height, such as ha<50nm, the mode area undergoes only small expansion, yet the propagation length could see a notable increase. For example, when ha = 10nm, wa = 150nm, the propagation length can reach ~35 μm, more than 2 times as large as that of the conventional DLSPP mode (~17 μm), while there is merely a less than 30% increase in its mode area. Therefore a shallow and wide nanohole could not only cause strong field enhancement as shown in Fig. 6, but also effectively reduce the transmission loss while maintaining deep-subwavelength confinement as shown in Fig. 7. At larger ha, the mode area increases rapidly as the width of the hole increases. In such cases, the greatly extended propagation length comes at a price of much larger mode sizes. When ha = 100nm or 150nm, the propagation distance could exceed 100 μm, but its normalized mode area also approaches 0.5. It is also noted that at rather large nanohole size, the FOM could decrease with increasing ha (e.g. FOM(wa = 150nm,ha = 150nm) < FOM(wa = 150nm,ha = 100nm)), as the increase of the mode area outruns the increase of the propagation length,
We note that there is a special case that deserves further discussions. When wa = 200nm, i.e. the width of nanohole equals that of the ridge, the air nanohole turns into a nano-gap and the DLSPP structure transforms into a hybrid plasmonic waveguide [15–17]. However, such configuration with a suspended silicon structure over an air gap on top of the metal surface is physically infeasible to implement and the air gap has to be replaced by a low-index dielectric buffer layer. Due to the higher index of the buffer layer, the field confinement effect would be weaker in that case compared to ours.
In fact, for practical applications, etching and further deposition may generate nanoholes with various geometric shapes (e.g. trapezoidal) and positions (e.g. non-concentric). Consequently it could result in different modal behaviors, which may be more complicated than what is studied here and deserve further investigation in our later studies.
In this paper, we have proposed and studied a novel DLSPP waveguide by simply incorporating an air nanohole in the dielectric ridge. The existence of a shallow and wide, low-index nanohole could cause strong field enhancement in the nanohole and thus significant changes in the modal behavior of the plasmonic mode. Reductions in the transmission loss could be achieved when the mode area is only slightly increased under such a design. Our proposed structure could provide a relatively simple and effective way to alleviate the loss problem in the plasmonic waveguides, and could be appealing for building ultra-compact plasmonic devices and high-density photonic integrated circuits.
This work was supported by 973 Program (2009CB930701), NSFC (60921001/61077064) and PCSIRT, SEM, and the Innovation Foundation of BUAA for PhD Graduates.
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