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In this article, we propose a novel kind of widely tunable surface plasmon polaritons (SPP) bandgap in a Kerr nonlinear metal-insulator-metal waveguide. By two identical gratings, the pump beam is coupled to two opposing SPP waves, which interfere with each other and results in SPP standing wave in the region between the two gratings. The refractive index of the Kerr nonlinear material is then periodically modulated by the SPP standing wave, and a SPP bandgap is formed. The position of the SPP bandgap can be tuned from 1.4 μm to 1.75 μm by adjusting the pump wavelength, and the relationship between the transmittance contrast of the bandgap and the pump power is also studied. Comparing with existing methods that directly modulate the refractive index (RI) or the width of the waveguide, in our work, the periodic modulation of the RI comes from the interference of the pump light, which can greatly simplify the fabrication. This work may find applications in the design of novel nonlinear devices for future all-optical integrated circuits.
© 2014 Optical Society of America
1. Introduction
Surface plasmon polariton (SPP) waveguides, which can provide the subwavelength-scale confinement of optical energy transmission, have been studied extensively in the last decade [1,2]. Various SPP waveguides, such as insulator-metal-insulator (IMI), metal-insulator-metal (MIM), channel plasmon polaritons (CPP),dielectric-loaded SPP waveguides (DLSPPW) and hybrid plasmonic waveguide, have been proposed theoretically and experimentally [3–6]. Among these, the MIM structure is particularly attractive due to its excellent mode-field confinement, acceptable SPP propagation length, easy fabrication, and efficient couplings with optical fiber and silicon waveguides [7, 8]. Nowadays, many MIM-based plasmonic devices, such as filters, splitters, wavelength division multiplexers (WDMs) and switches, have been demonstrated due to their potential applications in future ultracompact photonic integrated circuits (PIC) [9–11].
MIM waveguide Bragg gratings (MIM-WBGs), which may find applications in nanophotonics in a similar way to fiber Bragg gratings (FBGs) in classical optics, have received much attention in recent years [12,13]. Previous studies have shown that a MIM-WBG can be formed by periodically concatenating two MIM sections with different effective refractive indices (neff = β/k0, where β is the propagation constant of the SPP mode, and k0 is the free space wavenumber) along the propagation direction. And the effective index of a MIM section can be modified by changing the refractive index [14–16] or the width of the insulator layer [12,17,18]. This periodical modulation of the effective index leads to the formation of a bandgap in the transmittance spectrum, which can be used to realize the wavelength-sensitive operations. However, in recent years, two difficulties limit the development of the bandgap-based plasmonic devices. One is the complicated fabrication process [12,18]. For both the refractive index modulation and the width modulation cases, it requires very high fabrication precision to form the WBGs in a subwavelength insulator layer, so the fabrication repeatability is poor. And the other is the narrow wavelength tunability of the bandgap. As is known, the bandgap is defined once the Bragg grating structure is fabricated. Even if some Kerr nonlinear material is introduced to bring some tunability in the bandgap, the tuning wavelength range is usually as narrow as tens of nanometers due to the low third-order nonlinear coefficient of the nonlinear materials [19–22]. So, in this article, we propose a novel method of SPP interference to form the WBG in a nonlinear MIM waveguide. In this method, the periodical modulation of refractive index (RI) is origin from the SPP interference, which overcomes the fabrication difficulties of periodical micro-structures effectively; at the same time, the RI modulation period can be easily tuned by changing the pump wavelength, which results in a widely tunable SPP bandgap with one MIM structure.
2. Results and discussion
The MIM structure we studied is shown in Fig. 1(a). A Kerr-nonlinear layer (the blue region) and two identical gratings etched in the metal layer are introduced into a typical MIM waveguide. The gratings can couple the pump beam into SPP waves in the waveguide, and the SPP waves stimulated by left and right gratings will interfere with each other, which results in the SPP standing wave between the two gratings. Further, since the RI of Kerr-nonlinear material is linearly related to optical intensity (|E|2), the RI of the Kerr material in the MIM waveguide will be periodically modulated along the propagation direction. If the RI modulation contrast ratio is large enough, an SPP bandgap will take place. In the following, we will give the details.
Fig. 1 (a) The schematic diagram of the nonlinear MIM structure; (b) The electric field distribution of the MIM waveguide under plane-wave incidence without considering nonlinearity.
Without the loss of generality, we select Ag and dye doped poly(methyl methacrylate) (PMMA) as the metallic and Kerr-nonlinear materials in the MIM waveguide, respectively, and set the thickness of both the PMMA and Ag layers to 0.25 μm. The linear RI and third-order nonlinear coefficient of the dye doped PMMA are n0 = 1.49 and n2 = 5.7 × 10−10 cm2/W from Ref. [23] and the frequency-dependent RI of Ag is taken from Ref. [24]. A p-polarized pump light with the wavelength of 1.55 μm is incident normally onto the top of MIM waveguide [see Fig. 1(a)]. The left and right coupling gratings are identical, and the width and depth of the grooves are 0.2 μm and 0.22 μm, respectively. Here, we should point out that the groove depth influences the coupling efficiency greatly. Deeper grooves leads to higher coupling efficiency. However, in our designs, the coupling efficiency is not the higher the better. This is mainly because that except for coupling the pump light into the MIM waveguide, the gratings also couple part of the signal light outside the waveguide. So a moderate coupling efficiency should be adopted to make a balance between them, and the 0.22 μm depth is the optimized value.
Firstly, we determine the grating period to ensure the efficient stimulation of the SPP waves, which requires the momentum matching between the incident beam and the guided SPP mode. On one hand, the momentum supplied by the grating is 2πm/a, where m is the diffractive ordser. On the other hand, the propagation constant (β) of the guided SPP mode in MIM waveguide can be obtained by the following dispersion relation [25, 26]
where ε1 and ε2 are the dielectric constants of dielectric and metallic materials, wI is the thickness of dielectric layer, and k0, k1, k2 are the wavenumber in vacuum, dielectric and metal medium, respectively. Although this dispersion relation (Eqs. (1)–(2)) is derived assuming infinite thick metal layer, the thickness of the metal layer, in our case, is much larger than the penetration depth of SPP mode in the MIM waveguide. So this dispersion relation is still applicable. Here, in our case, substituting λ0 = 1.55 μm, wI = 0.25 μm, , ε2 = −114 + 11.51i to Eqs. ((1)–(2)), we get β = 6.59 + 0.00689i μm−1, thus the optimized grating period is a = 2π /β ≈ 0.95 μm.Secondly, with this optimized grating period, we study the interference properties of pump beam stimulated by the two identical gratings in the MIM waveguide with the finite-difference time-domain (FDTD) method. According to the dispersion relation Eqs. ((1)–(2)), the 3 dB propagation length at the incident wavelength of 1.55 μm is about 12 μm. So we set the distance between the two gratings to be 12 μm. Now we inspect the linear responds of the MIM structure. The normalized electric field (|E/E0|) in the MIM waveguide stimulated by the two gratings was calculated [see Fig. 1(b)]. The SPP standing wave can be obviously observed in the PMMA layer. The electric field intensity at the antinode is about 10 times larger than that at the node. This standing wave originates from the interference between the right propagating SPP wave from the left grating and the left propagating SPP wave from the right grating. We also notice that at the symmetric plane of two gratings (i.e. x = 0), there is a wave node, which means a π phase-difference between the left and right propagating SPP wave.
Now we consider the nonlinear property of the PMMA. Since the RI change is proportional to the local optical intensity (E2) for the Kerr nonlinear material, the RI of the PMMA will be sine-modulated. However, the change of the RI distribution will backwardly affect the propagation behavior of the SPP waves, and thus change the optical intensity distribution. When the optical intensity of the SPP standing wave just maintains the RI distribution in the waveguide, we call it a steady-state distribution. To solve the steady-state distribution for a specified pump wavelength and intensity, the iterative FDTD method is used:
where n(i) and E(i) are the RI and electric field distribution in the PMMA layer in the ith iteration, ndirect is the RI distribution directly calculated from the electric field distribution, and f is a factor between 0 and 1, which is used for avoiding the numerical divergence. In our simulations, the initial RI distribution is set to n0, and the convergence criteria is the average RI difference between ith and (i + 1)th iteration is small enough.Next, let us consider a pump beam with the wavelength of 1.55 μm and intensity of 40 MW/cm2. By using the iterative FDTD method, we obtain the steady-state electric field distribution, as shown in Fig. 2(a). The electric field is normalized to that of pump beam. It should be noticed that, the period of the SPP standing wave (0.229 μm) is shorter than that of the linear case (0.236 μm) since the average RI in the MIM waveguide is increased by the pump light. The calculated steady-state RI distribution in the MIM waveguide is shown in Fig. 2(b). Between the two gratings, a periodic RI modulation is observed. The maximal RI at the antinode is about 1.507, whose relative RI change is about 1%. This periodical modulated RI in a MIM waveguide will form a SPP bandgap. Here, we calculate the transmittance spectrum by the FDTD technique, and the result is shown in Fig. 2(c) with the black line. In the simulation, a broadband TM wave with magnetic field in the z-direction is used to excite the SPP modes in the waveguide, and a receiver plane is adopted to record the transmission power. The transmittance spectrum is obtained by comparing the transmission power with or without the periodical RI modulation. An obvious bandgap appears at the wavelengths ranges from 1.5 μm to 1.6 μm, and the transmittance contrast is nearly 50%, which is dependent on the pump intensity. Compared with FBGs, the bandgap is very wide, which may find application in wideband SPP bandstop filters or reflectors. We also calculated the transmittance spectra for the pump intensities of 20 MW/cm2, 60 MW/cm2, 100 MW/cm2, and the results are shown in Fig. 2(c). It can be clearly seen that when the pump beam is stronger, the transmittance contrast of the SPP bandgap is higher due to the deeper RI modulation in the PMMA layer. For the pump intensity of 100 MW/cm2, the contrast reaches 85%. In addition, the center wavelength of the bandgap blue shifts slightly for stronger pump intensity, because the larger average RI of MIM waveguide shortens the RI modulation period.
Fig. 2 The electric field distribution in the MIM waveguide under a 40 MW/cm2 pump beam at the wavelength of 1.55 μm; (b) The modulated RI in the MIM waveguide; (c) The transmittance spectra of the MIM waveguide for different pump powers from 20 to 100 MW/cm2.
In the above analyses, we only consider the case of 1.55 μm pump wavelength. To realize the bandgap tunability, the pump wavelength is varied. Here we first investigate the coupling efficiency. Since our coupling gratings only have five periods, we have a wide coupling line-width. We calculate the electric field intensities for the pump wavelengths ranging from 1.2 μm to 2.0 μm and pump power density of 100 MW/cm2 by the iterative FDTD method. By extracting the electric field intensity at the center-line [the blue dashed line in Fig. 2(a)] of the PMMA layer, we show the wavelength response in Fig. 3(a), and the positions of the grating area are also marked with blue lines for convenience. As is shown in the figure, the SPP standing wave can be efficiently stimulated for the pump wavelength between 1.55 μm and 1.7 μm [yellow area in Fig. 3(a)]. We can also notice that outside the standing wave region, there are some horizontal strips. When the length of the standing wave region is integer times of the SPP wavelength (or L = mλspp), the SPP wave stimulated by the left grating is in phase of that by the right grating, so the interference will be constructive outside the standing wave region, leading to a bright strip; when L = (m + 1/2)λspp, they will have opposite phases, so there are dark strips. It should be noticed that for the effectively stimulated wavelengths (1.55 μm to 1.7 μm), the intensity of the SPP standing wave is little affected, no matter how the wavelength corresponds to the bright or dark strip.
Fig. 3 (a) The electric field distribution in the center line of the MIM waveguide for wavelengths from 1.2 μm to 2.0 μm; (b) The transmittance spectra for pump wavelengths from 1.4 μm to 1.,75 μm, respectively. The pump power is 100 MW/cm2.
Further, we calculate the transmittance spectra for the pump wavelengths from 1.4 μm to 1.75 μm. As is shown in Fig. 3(b), the bandgap redshifts when the pump wavelength is longer. This redshift originates from the increased modulation period of the standing wave for longer pump wavelength [see Fig. 3(a)]. At the well coupled wavelength ranges (such as 1.55 μm, 1.60 μm and 1.65 μm), the transmittance contrast is high (reaches 85%). When the pump wavelength is far away from the 1.55 μm (where the coupling gratings are optimized to realize the maximal coupling efficiency at this wavelength), the transmittance contrast decreases greatly due to the weak coupling of the pump beam by the gratings. So, we have realized a widely tunable SPP bandgap from 1.4 to 1.8 μm at the same MIM waveguide by only changing the pump wavelengths, which provides a new way to design nonlinear devices in the MIM waveguide.
3. Conclusion
In conclusion, we have demonstrated the generation of SPP bandgap based on the RI sine modulation by the SPP standing wave in the nonlinear Ag-PMMA-Ag waveguide. The SPP standing wave comes from the interference between two opposing SPP waves, which are stimulated by two identical gratings in the Ag layer under plane-wave incidence. The standing wave can sine-modulate the RI of the Kerr-type PMMA layer, and cause the SPP bandgap. Our simulations reveal that the properties of SPP bandgap greatly depend on the pump intensity or wavelength. The pump intensity mainly influences the transmittance contrast, higher transmittance contrast is observed with stronger pump intensity; while the pump wavelength affects the wavelength of the SPP bandgap. The bandgap will red-shift with the longer pump wavelength. On this basis, we have realized a widely tunable SPP bandgap from 1.4 μm to 1.75 μm with the same MIM waveguide only changing the pump wavelengths. This work can overcome the difficulties of fabricating periodical nanostructure in the SPP waveguide, and may find applications in the design of novel nonlinear devices for future all-optical integrated circuits.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11104282, 11204317, 11374303).
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